Spectral Properties of Graphs

Abstract

In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. The set of eigenvalues of a graph G is known as the spectrum of G and denoted by Sp(G). We compute the spectra of some well-known families of graphs—the family of complete graphs, the family of cycles etc. We present Sachs’ theorem on the spectrum of the line graph of a regular graph. We also obtain the spectra of product graphs—Cartesian product, direct product, and strong product. We introduce Cayley graphs and Ramanujan graphs and highlight their importance. Finally, as an application of graph spectra to chemistry, we discuss the “energy of a graph”—a graph invariant that is widely studied these days. All graphs considered in this chapter are finite, undirected, and simple.

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11.1 Introduction

In this chapter, we look at the properties of graphs from our knowledge of their eigenvalues. The set of eigenvalues of a graph G is known as the spectrum of G and denoted by Sp(G). We compute the spectra of some well-known families of graphs—the family of complete graphs, the family of cycles etc. We present Sachs’ theorem on the spectrum of the line graph of a regular graph. We also obtain the spectra of product graphs—Cartesian product, direct product, and strong product. We introduce Cayley graphs and Ramanujan graphs and highlight their importance. Finally, as an application of graph spectra to chemistry, we discuss the “energy of a graph”—a graph invariant that is widely studied these days. All graphs considered in this chapter are finite, undirected, and simple.

11.2 The Spectrum of a Graph

Let G be a graph of order n with vertex set V = { v ₁, …, v _n }. The adjacency matrix of G (with respect to this labeling of V ) is the n by n matrix A = (a _ij ), where

$${a}_{ij} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\mathrm{if}\ {v}_{i}\ \mathrm{ is adjacent to}\ {v}_{j}\ \mathrm{in}\ G \\ 0\quad &\mathrm{otherwise.}\\ \quad \end{array} \right .$$

Thus A is a real symmetric matrix of order n. Hence,

• (i) The spectrum of A, that is, the set of its eigenvalues is real.

• (ii) ℝⁿ has an orthonormal basis of eigenvectors of A.

• (iii) The sum of the entries of the ith row (column) of A is d(v _i ) in G.

The spectrum of A is called the spectrum of G and denoted by Sp(G). We note that Sp(G), as defined above, depends on the labeling of the vertex set V of G. We now show that it is independent of the labeling of G. Suppose we consider a new labeling of V . Let A′ be the adjacency matrix of G with respect to this labeling. The new labeling can be obtained from the original labeling by means of a permutation π of V (G). Any such permutation can be effected by means of a permutation matrixP of order n (got by permuting the rows of I _n , the identity matrix of order n). (For example, if n = 3, the permutation matrix $P = \left (\begin{array}{ccc} 0&\,\,\,1&\,\,\,0\\ 0 &\,\,\,0 &\,\,\,1 \\ 1&\,\,\,0&\,\,\,0 \end{array} \right )$ takes v ₁, v ₂, v ₃ to v ₃, v ₁, v ₂ respectively since $\left (\begin{array}{ccc} 1&2&3\\ \end{array} \right )\,\left (\begin{array}{ccc} 0&\,\,\,1&\,\,\,0\\ 0 &\,\,\,0 &\,\,\,1 \\ 1&\,\,\,0&\,\,\,0 \end{array} \right ) = \left (\begin{array}{ccc} 3&1&2\\ \end{array} \right ).$

Let P = (p _ij ). Now given the new labeling of V , that is, given the permutation π on {1, 2, …, n}, and the vertices v _i and v _j , there exist unique α₀ and β₀, such that π(i) = α₀ and π(j) = β₀ or equivalently ${p}_{{\alpha }_{0}i} = 1$ and ${p}_{{\beta }_{0}j} = 1,$ while for α≠α₀ and β≠β₀, p _αi  = 0 = p _βj . Thus, the (α₀, β₀)th entry of the matrix A′ = PAP ^− 1 = PAP ^T (where P ^T stands for the transpose of P) is

$$\sum \limits_{k,l=1}^{n}{p}_{{ \alpha }_{0}k}{a}_{kl}{p}_{{\beta }_{0}l} = {a}_{ij}.$$

Hence, ${v}_{{\alpha }_{0}}{v}_{{\beta }_{0}} \in E(G)$ if and only if v _i v _j  ∈ E(G). This proves that the adjacency matrix of the same graph with respect to two different labelings are similar matrices. But then similar matrices have the same spectra. □

We usually arrange the eigenvalues of G in their nondecreasing order: λ₁ ≥ λ₂ ≥ … ≥ λ _n . If λ₁, …, λ _s are the distinct eigenvalues of G, and if m _i is the multiplicity of λ _i as an eigenvalue of G, we write

$$Sp(G) = \left (\begin{array}{cccc} {\lambda }_{1} & {\lambda }_{2} & \ldots & {\lambda }_{s}\\ {m}_{1 } & {m}_{2 } & \ldots &{m}_{s} \end{array} \right ).$$

Definition 11.2.1.

The characteristic polynomial of G is the characteristic polynomial of the adjacency matrix of G with respect to some labeling of G. It is denoted by χ(G; λ).

Hence, χ(G; λ) = det(xI − A) = det(P(xI − A)P ^− 1) = det(xI − PAP ^− 1) for any permutation matrix of P, and hence χ(G; λ) is also independent of the labeling of V (G).

Definition 11.2.2.

A circulant of order n is a square matrix of order n in which all the rows are obtainable by successive cyclic shifts of one of its rows (usually taken as the first row).

For example, the circulant with first row (a ₁  a ₂  a ₃) is the matrix $\left (\begin{array}{ccc} {a}_{1} & \,\,\,{a}_{2} & \,\,\,{a}_{3} \\ {a}_{3} & \,\,\,{a}_{1} & \,\,\,{a}_{2} \\ {a}_{2} & \,\,\,{a}_{3} & \,\,\,{a}_{1} \end{array} \right )$.

Lemma 11.2.3.

Let A be a circulant matrix of order n with first row (a ₁  a ₂ …a _n ). Then Sp(A) ={ a ₁ +a ₂ ω+⋯+a _n ω ⁿ⁻¹  : ω = an n th root of unity } ={ a ₁ +ζ ^r +ζ ^2r +⋯+ζ ^(n−1)r , 0 ≤ r ≤ n−1 and  ζ = a primitive n th root of unity }

Proof.

The characteristic polynomial of A is the determinant D = det(xI − A). Hence,

$$D = \left \vert \begin{array}{cccc} x - {a}_{1} & - {a}_{2} & \ldots & - {a}_{n} \\ - {a}_{n} &x - {a}_{1} & \ldots & - {a}_{n-1}\\ \vdots & \vdots & & \vdots \\ - {a}_{2} & - {a}_{3} & \ldots & x - {a}_{1} \end{array} \right \vert .$$

Let C _i denote the ith column of D, 1 ≤ i  ≤ n, and ω, an nth root of unity. Replace C ₁ by C ₁ + C ₂ω + ⋯ + C _n ω^n − 1. This does not change D. Let λ_ω = a ₁ + a ₂ω + ⋯ + a _n ω^n − 1. Then the new first column of D is (x − λ_ω, ω(x − λ_ω), …, ω^n − 1(x − λ_ω))^T, and hence x − λ_ω is a factor of D. This gives $D ={ \prod }_{\omega \,:\,{\omega }^{n}=1}(x - {\lambda }_{\omega })$, and Sp(A) = { λ_ω : ωⁿ = 1}. □

11.3 Spectrum of the Complete Graph K _n

For K _n , the adjacency matrix A is given by $A = \left (\begin{array}{ccccc} 0&\,\,\,1&\,\,\,1&\,\,\,\ldots &\,\,\,1\\ 1 &\,\,\,0 &\,\,\,1 &\,\,\,\ldots &\,\,\,1\\ \vdots & \,\,\,\vdots & \,\,\,\vdots & \,\,\, & \,\,\,\vdots \\ 1&\,\,\,1&\,\,\,1&\,\,\,\ldots &\,\,\,0\\ \end{array} \right )$, and so by Lemma 11.2.3,

$$\begin{array}{rcl}{ \lambda }_{\omega }& =& \omega + {\omega }^{2} + \cdots + {\omega }^{n-1} \\ & =& \left \{\begin{array}{l} n - 1\ \ \ \mathrm{if}\ \omega = 1\\ - 1\ \ \ \ \ \ \ \mathrm{if } \ \omega \neq 1. \end{array} \right. \end{array}.$$

Hence, $Sp({K}_{n}) = \left (\begin{array}{cc} n - 1& - 1\\ 1 &n - 1\\ \end{array} \right )$.

11.4 Spectrum of the Cycle C _n

Label the vertices of C _n as 0, 1, 2, …, n − 1 in this order. Then i is adjacent to i ± 1(mod n). Hence,

$$A = \left [\begin{array}{ccccccc} 0&\,\,\,1&\,\,\,0&\,\,\,0&\,\,\,\ldots &\,\,\,0&\,\,\,1\\ 1 &\,\,\,0 &\,\,\,1 &\,\,\,0 &\,\,\,\ldots &\,\,\,0 &\,\,\,0\\ \vdots & \,\,\,\vdots & \,\,\,\vdots & \,\,\,\vdots & \,\,\,\ddots & \,\,\,\vdots & \,\,\,\vdots \\ 1&\,\,\,0&\,\,\,0&\,\,\,0&\,\,\,\ldots &\,\,\,1&\,\,\,0\\ \end{array} \right ]$$

is the circulant with the first row (0 1 0 … 0 1). Again, by Lemma 11.2.3, Sp(C _n ) = { ω^r + ω^{r(n − 1)} : 0  ≤ r  ≤ n − 1, where ω isaprimitiventhrootofunity}. Taking $\omega =\cos \frac{2\pi } {n} + i\sin \frac{2\pi } {n}$, we get ${\lambda }_{r} = {\omega }^{r} + {\omega }^{r(n-1)} = (\cos \frac{2\pi r} {n} + i\sin \frac{2\pi r} {n} ) + (\cos \frac{2\pi r(n-1)} {n} + i\sin \frac{2\pi r(n-1)} {n} )$. This simplifies to the following:

• (i) If n is odd, $Sp({C}_{n}) = \left (\begin{array}{cccc} 2&2\cos \frac{2\pi } {n} &\ldots &2\cos \frac{(n-1)\pi } {n} \\ 1& 2 &\ldots & 2\\ \end{array} \right )$.

• (ii) If n is even, $Sp({C}_{n}) = \left (\begin{array}{ccccc} 2&2\cos \frac{2\pi } {n} &\ldots &2\cos \frac{(n-1)\pi } {n} & - 2\\ 1 & 2 &\ldots & 2 & 1\\ \end{array} \right )$.

11.4.1 Coefficients of the Characteristic Polynomial

Let G be a connected graph on n vertices, and let χ(G; x) = det(xI _n  − A) = x ⁿ + a ₁ x ^n − 1 + a ₂ x ^n − 2 + ⋯ + a _n be the characteristic polynomial of G. It is easy to check that ( − 1)^r a _r  = sum of the principal minors of A of order r. (Recall that a principal minor of order r of A is the determinant minor of A common to the same set of r rows and columns.)

Lemma 11.4.1.

Let G be a graph of order n and size m, and let χ(G;x) = x ⁿ + a ₁ x ⁿ⁻¹ + a ₂ x ⁿ⁻² + ⋯ + a _n be the characteristic polynomial of A. Then

• (i) a ₁ = 0

• (ii) a ₂ = −m

• (iii) a ₃ = − (twice the number of triangles in G)

Proof.

• (i) Follows from the fact that all the entries of the principal diagonal of A are zero.

• (ii) A nonvanishing principal minor of order 2 of A is of the form $\left \vert \begin{array}{cc} 0&\,\,\,1\\ 1 &\,\,\,0\\ \end{array} \right \vert$ and its value is − 1. Since any 1 in A corresponds to an edge of G, we get (ii).

• (iii) A nontrivial principal minor of order 3 of A can be one of the following three types:

  $$\left \vert \begin{array}{ccc} 0&\,\,\,1&\,\,\,0\\ 1 &\,\,\,0 &\,\,\,0 \\ 0&\,\,\,0&\,\,\,0\\ \end{array} \right \vert ,\left \vert \begin{array}{ccc} 0&\,\,\,1&\,\,\,1\\ 1 &\,\,\,0 &\,\,\,0 \\ 1&\,\,\,0&\,\,\,0\\ \end{array} \right \vert ,\left \vert \begin{array}{ccc} 0&1&1\\ 1 &0 &1 \\ 1&1&0\\ \end{array} \right \vert .$$

  Of these, only the last determinant is nonvanishing. Its value is 2 and corresponds to a triangle in G. This proves (iii).□

11.5 The Spectra of Regular Graphs

In this section, we look at the spectra of some regular graphs.

Theorem 11.5.1.

Let G be a k-regular graph of order n. Then

• (i) k is an eigenvalue of G.

• (ii) If G is connected, every eigenvector corresponding to the eigenvalue k is a multiple of 1, (the all 1-column vector of length n) and the multiplicity of k as an eigenvalue of G is one.

• (iii) For any eigenvalue λ of G, |λ|≤ k. (Hence Sp(G) ⊂ [−k,k]).

Proof.

• (i) We have A 1 = k 1, and hence k is an eigenvalue of A.

• (ii) Let x = (x ₁, …, x _n )^T be any eigenvector of A corresponding to the eigenvalue k so that A x = k x. We may suppose that x has a positive entry (otherwise, take − x in place of x) and that x _j is the largest positive entry in x. Let ${v}_{{i}_{1}},{v}_{{i}_{2}},\ldots ,{v}_{{i}_{k}}$ be the k neighbors of v _j in G. Taking the inner product of the jth row of A with x, we get ${x}_{{i}_{1}} + {x}_{{i}_{2}} + \cdots + {x}_{{i}_{k}} = k{x}_{j}$. This gives, by the choice of x _j , ${x}_{{i}_{1}} = {x}_{{i}_{2}} = \ldots = {x}_{{i}_{k}} = {x}_{j}$.

  Now start at ${v}_{{i}_{1}},{v}_{{i}_{2}},\ldots ,{v}_{{i}_{k}}$ in succession and look at their neighbors in G. As before, the entries x _p in x corresponding to these neighbors must all be equal to x _j . As G is connected, all the vertices of G are reachable in this way step by step. Hence x = x _j (1, 1, …, 1)^T, and every eigenvector x of A corresponding to the eigenvalue k is a multiple of 1. Thus, the space of eigenvectors of A corresponding to the eigenvalue k is one-dimensional, and therefore, the multiplicity of k as an eigenvalue of G is 1.

• (iii) The proof is similar to (ii). If A y = λy, y≠0, and if y _j is the entry in y with the largest absolute value, we see that the equation ${\sum \nolimits }_{p=1}^{k}{y}_{{i}_{p}} = \lambda {y}_{j}$ implies that $\vert \lambda \vert \vert {y}_{j}\vert = \vert \lambda {y}_{j}\vert = {\vert }{\sum }_{p=1}^{k}{y}_{{i}_{p}}{\vert }\leq {\sum }_{p=1}^{k}\vert {y}_{{i}_{p}}\vert \,\leq \, k\vert {y}_{j}\vert$. Thus, | λ |  ≤ k. □

Corollary 11.5.2.

If Δ denotes the maximum degree of G, then for any eigenvalue λ of G, |λ|≤ Δ.

Proof.

Considering a vertex v _j of maximum degree Δ, and imitating the proof of (iii) above, we get | λ | | y _j  | ≤ Δ | y _j  | . □

11.5.1 The Spectrum of the Complement of a Regular Graph

Theorem 11.5.3.

Let G be a k-regular connected graph of order n with spectrum $\left (\begin{array}{ccccc} k& {\lambda }_{2} & {\lambda }_{3} & \ldots & {\lambda }_{s} \\ 1 &{m}_{2} & {m}_{3} & \ldots &{m}_{s}\\ \end{array} \right )$ . Then the spectrum of G ^c , the complement of G, is given by $Sp({G}^{c}) = \left (\begin{array}{ccccc} n - 1 - k & - {\lambda }_{2} - 1 & - {\lambda }_{3} - 1 & \ldots & - {\lambda }_{s} - 1 \\ 1 & {m}_{2} & {m}_{3} & \ldots & {m}_{s}\\ \end{array} \right )$ .

Proof.

As G is k-regular, G ^c is n − 1 − k regular, and hence by Theorem 11.5.1, n − 1 − k is an eigenvalue of G ^c. Further, the adjacency matrix of G ^c is A ^c = J − I − A, where J is the all-1 matrix of order n, I is the identity matrix of order n, and A is the adjacency matrix of G. If χ(λ) is the characteristic polynomial of A,  χ(λ) = (λ − k)χ₁(λ). By Cayley–Hamilton theorem, χ(A) = 0 and hence we have Aχ₁(A) = kχ₁(A). Hence, every column vector of χ₁(A) is an eigenvector of A corresponding to the eigenvalue k. But by Theorem 11.5.1, the space of eigenvectors of A is generated by 1, G being connected. Hence, each column vector of χ₁(A) is a multiple of 1. But χ₁(A) is symmetric and hence χ₁(A) is a multiple of J, say, χ₁(A) = αJ, α≠0. Thus, J and hence J − I − A are polynomials in A (remember: A ⁰ = I). Let λ≠k be any eigenvalue of A [so that χ₁(λ) = 0], and Y an eigenvector of A corresponding to λ. Then

$$\begin{array}{rcl}{ A}^{c}Y & =& (J - I - A)Y \\ & =& \left(\frac{{\chi }_{1}(A)} {\alpha } - I - A\right)Y \\ & =& \left(\frac{{\chi }_{1}(\lambda )} {\alpha } - 1 - \lambda \right)Y \ \text{ (see Note 11.5.4 below)} \\ & =& (-1 - \lambda)Y\end{array}.$$

Thus, A ^c Y = ( − 1 − λ)Y , and therefore − 1 − λ is an eigenvalue of A ^c corresponding to the eigenvalue λ(≠k) of A. □

Note 11.5.4.

We recall that if f(λ) is a polynomial in λ, and $Sp(A) = \left (\begin{array}{cccc} {\lambda }_{1} & {\lambda }_{2} & \ldots & {\lambda }_{s}\\ {m}_{1 } & {m}_{2 } & \ldots &{m}_{s}\\ \end{array} \right )$, then $Sp(f(A)) = \left (\begin{array}{cccc} f({\lambda }_{1})&f({\lambda }_{2})&\ldots &f({\lambda }_{s})\\ {m}_{ 1} & {m}_{2} & \ldots & {m}_{s}\\ \end{array} \right )$.

11.5.2 Spectra of Line Graphs of Regular Graphs

We now establish Sachs’ theorem, which determines the spectrum of the line graph of a regular graph G in terms of Sp(G).

Let G be a labeled graph with vertex set V (G) = { v ₁, …, v _n } and edge set E(G) = { e ₁, …, e _m }. With respect to these labelings, the incidence matrix B = (b _ij ) of G, which describes the incidence structure of G, is defined as the m by n matrix B = (b _ij ), where ${b}_{ij} = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\mathrm{if}\ {e}_{i}\ \mathrm{ is incident to}\ {v}_{j}, \\ 0\quad &\mathrm{otherwise.} \end{array} \right .$

Lemma 11.5.5.

Let G be a graph of order n and size m with A and B as its adjacency and incidence matrices, respectively. Let A _L denote the adjacency matrix of the line graph of G. Then

• (i) BB ^T = A _L + 2I _m .

• (ii) If G is k-regular, B ^T B = A + kI _n .

Proof.

Let A = (a _ij ) and B = (b _ij ). We have

• (i)
  $$\begin{array}{lll}{ (B{B}^{\mathrm{T}})}_{ ij}& ={ \sum }_{p=1}^{n}{b}_{ip}{b}_{jp} & \\ & = \text{ number of vertices ${v}_{p}$ that are incident to both ${e}_{i}$ and ${e}_{j}$} & \\ & = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\quad &\mathrm{if}\ {e}_{i}\ \mathrm{and}\ {e}_{j}\ \mathrm{ are adjacent} \\ 0\quad &\mathrm{if}\ i\neq j\ \mathrm{and}\ {e}_{i}\ \mathrm{and}\ {e}_{j}\ \mathrm{ are nonadjacent} \\ 2\quad &\mathrm{if}\ i = j.\end{array} \right .& \\ \end{array}$$

• (ii) Proof of (ii) is similar. □

Theorem 11.5.6 (Sachs’ theorem). 

Let G be a k-regular graph of order n. Then χ(L(G);λ) = (λ + 2) ^m−n χ(G;λ + 2 − k), where L(G) is the line graph (see  Chap. 1 ) of G.

Proof.

Consider the two partitioned matrices U and V , each of order n + m (where B stands for the incidence matrix of G):

$$U = \left [\begin{array}{cc} \lambda {I}_{n}& - {B}^{\mathrm{T}} \\ 0 & {I}_{m}\\ \end{array} \right ],\,V = \left [\begin{array}{cc} {I}_{n}& {B}^{\mathrm{T}} \\ B &\lambda {I}_{m}\\ \end{array} \right ].$$

We have

$$UV = \left [\begin{array}{cc} \lambda {I}_{n} - {B}^{\mathrm{T}}B & 0 \\ B &\lambda {I}_{m}\\ \end{array} \right ]\ \mathrm{and}\ V U = \left [\begin{array}{cc} \lambda {I}_{n}& 0 \\ \lambda B &\lambda {I}_{m} - B{B}^{\mathrm{T}}\\ \end{array} \right ].$$

Now det(UV ) = det(VU) gives:

$${\lambda }^{m}\ \mathrm{det}(\lambda {I}_{ n} - {B}^{\mathrm{T}}B) = {\lambda }^{n}\ \mathrm{det}(\lambda {I}_{ m} - B{B}^{\mathrm{T}}).$$

(11.1)

Replacement of λ by λ + 2 in (11.1) yields

$${(\lambda + 2)}^{m-n}\ \mathrm{det}((\lambda + 2){I}_{ n} - {B}^{\mathrm{T}}B) = \mathrm{det}((\lambda + 2){I}_{ m} - B{B}^{\mathrm{T}}).$$

(11.2)

Hence, by Lemma 11.5.5,

$$\begin{array}{lllll} \chi (L(G);\lambda )& = \mathrm{det}(\lambda {I}_{m} - {A}_{L}) & \\ & = \mathrm{det}((\lambda + 2){I}_{m} - ({A}_{L} + 2{I}_{m})) & \\ & = \mathrm{det}((\lambda + 2){I}_{m} - B{B}^{\mathrm{T}}) & \\ & = {(\lambda + 2)}^{m-n}\ \mathrm{det}((\lambda + 2){I}_{n} - {B}^{\mathrm{T}}B)\quad \text{ (by (11.2))} & \\ & = {(\lambda + 2)}^{m-n}\ \mathrm{det}((\lambda + 2){I}_{n} - (A + k{I}_{n}))\quad \text{ (by Lemma 11.5.5)}& \\ & = {(\lambda + 2)}^{m-n}\ \mathrm{det}((\lambda + 2 - k){I}_{n} - A) & \\ & = {(\lambda + 2)}^{m-n}\ \chi (G;\lambda + 2 - k). & \\ \end{array}$$

□

Sachs’ theorem implies the following: As χ(G; λ) = $\prod \limits_1^n (\lambda - {\lambda_i})$, it follows that

$$\begin{array}{rcl} \chi (L(G);\lambda )& = {(\lambda + 2)}^{m-n}\ { \prod }_{i=1}^{n}(\lambda + 2 - k - {\lambda }_{i}) & \\ & = {(\lambda + 2)}^{m-n}\ { \prod }_{i=1}^{n}{(}\lambda - (k - 2 + {\lambda }_{i}){)}.& \\ \end{array}$$

Hence if $Sp(G) = \left (\begin{array}{cccc} k& \,\,\,{\lambda }_{2} & \,\,\,\ldots & \,\,\,{\lambda }_{s} \\ 1 &\,\,\,{m}_{2} & \,\,\,\ldots &\,\,\,{m}_{s}\\ \end{array} \right )$, then

$$Sp(L(G)) = \left (\begin{array}{ccccc} 2k - 2&\,\,\,k - 2 + {\lambda }_{2} & \,\,\,\ldots &\,\,\,k - 2 + {\lambda }_{s}& \,\,\, - 2 \\ 1 & \,\,\,{m}_{2} & \,\,\,\ldots & \,\,\,{m}_{s} &\,\,\,m - n \end{array} \right ).$$

We note that the spectrum of the Petersen graph can be obtained by using Theorems 11.5.3 and 11.5.6 (see Exercise 14.14). We use this result to prove a well-known result on the Petersen graph.

Theorem 11.5.7.

The complete graph K ₁₀ cannot be decomposed into (that is, expressed as an edge-disjoint union of) three copies of the Petersen graph.

Proof (Schwenk and Lossers [169]). 

Assume the contrary. Suppose K ₁₀ is expressible as an edge-disjoint union of three copies, say, P ₁, P ₂, P ₃, of the Petersen graph P (see Fig. 11.1). Then

$$A({K}_{10}) = J - I = A({P}_{1}) + A({P}_{2}) + A({P}_{3}),$$

(11.3)

Fig. 11.1

Petersen graph P

where A(H) stands for the adjacency matrix of the graph H, and J the all-1 matrix of order 10. (Note that each P _i is a spanning subgraph of K ₁₀, that the number of edges of P _i , namely 15, is a divisor of the number of edges of K ₁₀, namely, 45, and that the degree of any vertex v of P, namely, 3, is a divisor of the degree of v in K ₁₀, namely 9.) It is easy to check that 1 is an eigenvalue of P. Further, as P is 3-regular, the all-1 column vector 1 is an eigenvector of P. Now ℝ ¹⁰ has an orthonormal basis of eigenvectors of A(P) containing 1. Again, the null space of (A(P) − 1. I) = A(P) − I is of dimension 5 and is orthogonal to 1. (For P with the labeling of Fig. 11.1, one can check that for A(P) − I, the null space is spanned by the five vectors: (1 0 0 0 0 1  − 1 0 0  − 1)^T, (0 1 0 0 0  − 1 1  − 1 0 0)^T, (0 0 1 0 0 0  − 1 1  − 1 0)^T, (0 0 0 1 0 0 0 −1 1 −1)^T, (0 0 0 0 1  − 1 0 0  − 1 1)^T.) The orthogonal complement of 1 in ℝ ¹⁰ is of dimension 10 − 1 = 9. Hence, the null spaces of A(P ₁) − I and A(P ₂) − I must have a common eigenvector x orthogonal to 1. Multiplying the matrices on the two sides of (11.3) to x, we get (J − I)x = A(P ₁)x + A(P ₂)x + A(P ₃)x, that is (as J x = 0), − x = x + x + A(P ₃)x. Thus, A(P ₃)x =  − 3x, and this means that − 3 is an eigenvalue of the Petersen graph, a contradiction (see Exercise 14.14). □

Various proofs of Theorem 11.5.7 are available in literature. For a second proof, see [32].

11.6 Spectrum of the Complete Bipartite Graph K _p, q

We now determine the spectrum of the complete bipartite graph K _p, q .

Theorem 11.6.1.

$Sp({K}_{p,q}) = \left (\begin{array}{ccc} 0 &\sqrt{pq}& -\sqrt{pq}\\ p + q - 2 & 1 & 1\\ \end{array} \right )$ .

Proof.

Let V (K _p, q ) have the bipartition (X, Y ) with | X |  = p and | Y |  = q. Then the adjacency matrix of K _p, q is of the form

$$A = \left (\begin{array}{cc} 0 &{J}_{p,q} \\ {J}_{q,p}& 0\\ \end{array} \right ),$$

where J _r, s stands for the all-1 matrix of size r by s. Clearly, rank(A) = 2, as the maximum number of independent rows of A is 2. Hence, zero is an eigenvalue of A repeated p + q − 2 times (as the null space of A is of dimension p + q − 2). Thus, the characteristic polynomial ofA is of the form λ^{p + q − 2}(λ² + c ₂).

[Recall that by Lemma 11.4.1, the coefficient of λ^{p + q − 1} in χ(G; λ) is zero.] Further, again by the same lemma, − c ₂ = the number of edges of K _p, q  = pq. This proves the result.□

11.7 The Determinant of the Adjacency Matrix of a Graph

We now present the elegant formula given by Harary for the determinant of the adjacency matrix of a graph in terms of certain of its subgraphs.

Definition 11.7.1.

A linear subgraph of a graph G is a subgraph of G whose components are single edges or cycles.

Theorem 11.7.2 (Harary [92]). 

Let A be the adjacency matrix of a simple graph G. Then

$$\det A ={ \sum }_{H}{(-1)}^{e(H)}{2}^{c(H)},$$

where the summation is over all the spanning linear subgraphs H of G, and e(H) and c(H) denote, respectively, the number of even components and the number of cycles in H.

Proof.

Let G be of order n with V = { v ₁, …, v _n }, and A = (a _ij ). A typical term in the expansion of detA is

$$\mathit{sgn}(\pi ){a}_{1\pi (1)}{a}_{2\pi (2)}\ldots {a}_{n\pi (n)},$$

where π is a permutation on {1, 2, …, n} and sgn(π) = 1 or − 1 according to whether π is an even or odd permutation. This term is zero if and only if for some i, 1 ≤ i ≤ n, a _iπ(i) = 0, that is, if and only if π(i) = i or π(i) = j(≠i) and v _i v _j ∉E(G). Hence, this term is nonzero if and only if the permutation π is a product of disjoint cycles of length at least 2, and in this case, the value of the term is sgn(π). 1. 1…1 = sgn(π). Each cycle (ij) of length 2 in π corresponds to the single edge v _i v _j of G, while each cycle (ij…p) of length r > 2 in π corresponds to a cycle of length r of G. Thus, each nonvanishing term in the expansion of det(A) gives rise to a linear subgraph H of G and conversely. Now for any cycle C of S _n , sgn(C) = 1 or − 1 according to whether C is an odd or even cycle. Hence, sgn(π) = ( − 1)^e(H), where e(H) is the number of even components of H (that is, components that are either single edges or even cycles of the graph H). Moreover, any cycle of H has two different orientations. Hence, each of the undirected cycles of H of length ≥ 3 yields two distinct even cycles in S _n . [For example, the 4-cycle $({v}_{{i}_{1}}{v}_{{i}_{2}}{v}_{{i}_{3}}{v}_{{i}_{4}})$ gives rise to two cycles $({v}_{{i}_{1}}{v}_{{i}_{2}}{v}_{{i}_{3}}{v}_{{i}_{4}})$ and $({v}_{{i}_{4}}{v}_{{i}_{3}}{v}_{{i}_{2}}{v}_{{i}_{1}})$ in H.] This proves the result. □

Corollary 11.7.3 (Sachs [167]). 

Let χ(G;x) = x ⁿ + a ₁ x ⁿ⁻¹ + ⋯ + a _n be the characteristic polynomial of G. Then

$${a}_{i} ={ \sum }_{H}{(-1)}^{\omega (H)}{2}^{c(H)},$$

where the summation is over all linear subgraphs H of order i of G, and ω(H) and c(H) denote, respectively, the number of components and the number of cycle components of H.

Proof.

Recall that a _i  = ( − 1)ⁱ ∑ _H det(A), where H runs through all the induced subgraphs of order i of G. But by Theorem 11.7.2,

$$\mathrm{det}(A) ={ \sum }_{{H}_{i}}{(-1)}^{e({H}_{i})}{2}^{c({H}_{i})},$$

where H _i is a spanning linear subgraph of H and e(H _i ) stands for the number of even components of H _i , while c(H _i ) stands for the number of cycles in H _i . The corollary follows from the fact that i and the number of odd components of H _i have the same parity.□

11.8 Spectra of Product Graphs

We have already defined in Chap. 1 the graph products: Cartesian product, direct product, and strong product. In this section we determine the spectra of these graphs in terms of the spectra of their factor graphs. Our approach is based on Cvetković [46] as described in [47]. Recall that all these three products are associative.

Let $\mathcal{B}$ be a set of binary n-tuples (β₁, β₂, …, β _n ) not containing (0, 0, …, 0).

Definition 11.8.1.

Given a sequence of graphs G ₁, G ₂, …, G _n , the NEPS (Non-complete Extended P-Sum) of G ₁, G ₂, …, G _n , with respect to $\mathcal{B}$ is the graph G with V (G) = V (G ₁) ×V (G ₂) ×… ×V (G _n ), and in which two vertices (x ₁, x ₂, …, x _n ) and (y ₁, y ₂, …, y _n ) are adjacent if and only if there exists an n-tuple $({\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{n}) \in \mathcal{B}$ with the property that if β _i  = 1, then x _i y _i  ∈ E(G _i ) and if β _i  = 0, then x _i  = y _i .

Example 11.8.2.

• (i) n = 2 and $\mathcal{B} =\{ (1,1)\}$. Here the graphs are G ₁ and G ₂. The vertices (x ₁, x ₂) and (y ₁, y ₂) are adjacent in the NEPS of G ₁ and G ₂ with respect to $\mathcal{B}$ if and only if x ₁ y ₁ ∈ E(G ₁) and x ₂ y ₂ ∈ E(G ₂). Hence, G = G ₁ ×G ₂, the direct product of G ₁ and G ₂.

• (ii) n = 2 and $\mathcal{B} =\{ (0,1),(1,0)\}$. Here G is the Cartesian product G ₁ □G ₂.

• (iii) n = 2 and $\mathcal{B} =\{ (0,1),(1,0),(1,1)\}$. Here G = (G ₁ □G ₂) ∪(G ₁ ×G ₂) = G ₁ ⊠ G ₂, the strong product of G ₁ and G ₂.

Now, given the adjacency matrices of G ₁, …G _n , the adjacency matrix of the NEPS graph G with respect to the basis $\mathcal{B}$ is expressible in terms of the Kronecker product of matrices, which we now define:

Definition 11.8.3.

Let A = (a _ij ) be an m by n matrix and B = (b _ij ) be a p by q matrix. Then A ⊗ B, the Kronecker product of A with B, is the mp by nq matrix obtained by replacing each entry a _ij of A by the double array a _ij B (where a _ij B is the matrix obtained by multiplying each entry of B by a _ij ).

It is well known and easy to check that

$$(A \otimes B)(C \otimes D) = (AC \otimes BD),$$

(11.4)

whenever the matrix products AC and BD are defined. Clearly, this can be extended to any finite product whenever the products are defined.

Remark 11.8.4.

Let us look more closely at the product A ₁ ⊗ A ₂, where A ₁ and A ₂ are the adjacency matrices of the graphs G ₁ and G ₂ of orders n and t, respectively. To fix any particular entry of A ₁ ⊗ A ₂, let us first label V (G ₁) = V ₁ and V (G ₂) = V ₂ as V ₁ = { u ₁, …, u _n }, and V ₂ = { v ₁, …, v _t }. Then to fix the entry in A ₁ ⊗ A ₂ corresponding to ((u _i , u _j ), (v _p , v _q )), we look at the double array ${({A}_{1})}_{({u}_{i}{u}_{j})}{A}_{2}$ in A ₁ ⊗ A ₂, where ${({A}_{1})}_{({u}_{i}{u}_{j})} :=\, \alpha$ stands for the (i, j)th entry of A ₁. Then the required entry is just α{(p, q){ th entry of} A ₂}. Hence, it is 1 if and only if ${({A}_{1})}_{({u}_{i}{u}_{j})} = 1 = {({A}_{2})}_{({v}_{p}{v}_{q})}$, that is, if and only if u _i u _j  ∈ E(G ₁) and v _p v _q  ∈ E(G ₂), and 0 otherwise. In other words, A ₁ ⊗ A ₂ is the adjacency matrix of G ₁ ×G ₂. By associativity, A(G ₁) ⊗ … ⊗ A(G _r ) is the adjacency matrix of the graph product G ₁ ×… ×G _r .

Our next theorem determines the adjacency matrix of the NEPS G in terms of the adjacency matrices of G _i , 1 ≤ i ≤ n for all three products mentioned above.

Theorem 11.8.5 (Cvetković [46]). 

Let G be the NEPS of the graphs G ₁ ,…,G _n with respect to the basis $\mathcal{B}$ . Let A _i be the adjacency matrix of G _i , 1 ≤ i ≤ n. Then the adjacency matrix A of G is given by

$$A ={ \sum }_{\beta =({\beta }_{1},\ldots ,{\beta }_{n})\in \mathcal{B}}{A}_{1}^{{\beta }_{1} } \otimes \ldots \otimes {A}_{n}^{{\beta }_{n} }.$$

Proof.

Label the vertex set of each of the graphs G _i , 1 ≤ i ≤ n, and order the vertices of G lexicographically. Form the adjacency matrix A of G with respect to this ordering. Then (by the description of Kronecker product of matrices given in Remark 11.8.4) we have ${(A)}_{({x}_{1},\ldots ,{x}_{n})({y}_{1},\ldots ,{y}_{n})} ={ \sum }_{\beta \in \mathcal{B}}{({A}_{1}^{{\beta }_{1}})}_{({x}_{ 1},{y}_{1})}\ldots {({A}_{n}^{{\beta }_{n}})}_{ ({x}_{n},{y}_{n})}$, where (M)_(x, y) stands for the entry in M corresponding to the vertices x and y. But by lexicographic ordering, ${(M)}_{({x}_{1},\ldots ,{x}_{n})({y}_{1},\ldots ,{y}_{n})} = 1$ if and only if there exists a $\beta = ({\beta }_{1},\ldots ,{\beta }_{n}) \in \mathcal{B}$ with ${({A}_{i}^{{\beta }_{i}})}_{({x}_{ i},{y}_{i})} = 1$ for each i = 1, …, n. This of course means that x _i y _i  ∈ E(G _i ) if β _i  = 1 and x _i  = y _i if β _i  = 0 (the latter condition corresponds to ${A}_{i}^{{\beta }_{i}} = I$). □

We now determine the spectrum of the NEPS graph G with respect to the basis $\mathcal{B}$ in terms of the spectra of the factor graphs G _i .

Theorem 11.8.6 (Cvetković [46]). 

Let G be the NEPS of the graphs G ₁ ,…,G _n with respect to the basis $\mathcal{B}$ . Let k _i be the order of G _i and A _i , the adjacency matrix of G _i . Let $\{{\lambda }_{i1},\ldots ,{\lambda }_{i{k}_{i}}\}$ be the spectrum of G _i , 1 ≤ i ≤ n. Then

$$Sp(G) =\{ {\Lambda }_{{i}_{1}{i}_{2}\ldots {i}_{n}} : 1 \leq {i}_{j} \leq {k}_{j}\ \mathrm{and}\ 1 \leq j \leq n\},$$

where ${\Lambda }_{{i}_{1}{i}_{2}\ldots {i}_{n}} ={ \sum }_{\beta =({\beta }_{1},\ldots ,{\beta }_{n})\in \mathcal{B}}{\lambda }_{1{i}_{1}}^{{\beta }_{1}}\ldots ,{\lambda }_{n{i}_{ n}}^{{\beta }_{n}},\,1 \leq {i}_{ j} \leq {k}_{j}\ \mathrm{and}\ 1 \leq j \leq n$ .

Proof.

There exist vectors x _ij with A _i x _ij  = λ _ij x _ij , 1 ≤ i ≤ n; 1 ≤ j ≤ k _j . Now consider the vector $\text{ x} = {x}_{1{i}_{1}} \otimes \ldots \otimes {x}_{n{i}_{n}}$. Let A be the adjacency matrix of G. Then from Theorem 11.8.5 and (11.4) (rather its extension),

$$\begin{array}{rcl} A\text{ x}& =& \left ({\sum }_{\beta \in \mathcal{B}}{A}_{1}^{{\beta }_{1} } \otimes \ldots \otimes {A}_{n}^{{\beta }_{n} }\right )({x}_{1{i}_{1}} \otimes \ldots \otimes {x}_{n{i}_{n}}) \\ & =& {\sum }_{\beta \in \mathcal{B}}({A}_{1}^{{\beta }_{1} }{x}_{1{i}_{1}} \otimes \ldots \otimes {A}_{n}^{{\beta }_{n} }{x}_{n{i}_{n}}) \\ & =& {\sum }_{\beta \in \mathcal{B}}({\lambda }_{1{i}_{1}}^{{\beta }_{1} }{x}_{1{i}_{1}} \otimes \ldots \otimes {\lambda }_{n{i}_{n}}^{{\beta }_{n} }{x}_{n{i}_{n}}) \\ & =& \left ({\sum }_{\beta \in \mathcal{B}}{\lambda }_{1{i}_{1}}^{{\beta }_{1} }\ldots {\lambda }_{n{i}_{n}}^{{\beta }_{n} }\right )\text{ x} \\ & =& {\Lambda }_{{i}_{1}{i}_{2}\ldots {i}_{n}}\text{ x}\end{array}.$$

Thus, ${\Lambda }_{{i}_{1}{i}_{2}\ldots {i}_{n}}$ is an eigenvalue of G. This yields k ₁ k ₂ …k _n eigenvalues of G and hence all the eigenvalues of G. □

Corollary 11.8.7.

Let Sp(G ₁ ) ={ λ ₁ ,…,λ _n } and Sp(G ₂ ) ={ μ ₁ ,…,μ _t } and let A ₁ and A ₂ be the adjacency matrices of G ₁ and G ₂ , respectively. Then

• (i) A(G ₁ × G ₂ ) = A ₁ ⊗ A ₂ ; and Sp(G ₁ × G ₂ ) ={ λ _i μ _j  : 1 ≤ i ≤ n, 1 ≤ j ≤ t}.

• (ii) A(G ₁ □G ₂ ) = (I _n ⊗ A ₂ ) + (A ₁ ⊗ I _t ); and Sp(G ₁ □G ₂ ) ={ λ _i + μ _j  : 1 ≤ i ≤ n, 1 ≤ j ≤ t}.

• (iii) A(G ₁ ⊠ G ₂ ) = (A ₁ ⊗ A ₂ ) + (I _n ⊗ A ₂ ) + (A ₁ ⊗ I _t ); and Sp(G ₁ ⊠ G ₂ ) ={ λ _i μ _j + λ _i + μ _j  : 1 ≤ i ≤ n, 1 ≤ j ≤ t}.

□

11.9 Cayley Graphs

11.9.1 Introduction

Cayley graphs are a special type of regular graphs constructed out of groups.

Let Γ be a finite group and S ⊂ Γ be such that

• (i) e∉S (e is the identity of Γ),

• (ii) If a ∈ S, then a ^− 1 ∈ S, and

• (iii) S generates Γ.

Construct a graph G with V (G) = Γ and in which ab ∈ E(G) if and only if b = as for some s ∈ S. Since as = as′ in Γ implies that s = s′, it follows that each vertex a of G is of degree | S | ; that is, G is a | S | -regular graph. Moreover, if a and b are any two vertices of G, there exists c in Γ such that ac = b. But as S generates Γ, c = s ₁ s ₂ …s _p , where s _i  ∈ S, 1 ≤ i ≤ p. Hence, b = as ₁ s ₂ …s _p , which implies that b is reachable from a in G by means of the path a(as ₁)(as ₁ s ₂)…(as ₁ s ₂ …s _p  = b). Thus, G is a connected simple graph. [Condition (i) implies that G has no loops.] G is known as the Cayley graph Cay(Γ; S) of the group Γ defined by the set S.

We now consider a special family of Cayley graphs. Take Γ = (ℤ _n , + ), the additive group of integers modulo n. If S = { s ₁, s ₂, …, s _p }, then 0∉S and s _i  ∈ S if and only if n − s _i  ∈ S. The vertices adjacent to 0 are s ₁, s ₂, …, s _p , while those adjacent to i are (s ₁ + i)(mod n), (s ₂ + i)(mod n), …, (s _p  + i)(mod n). Consequently, the adjacency matrix of Cay(ℤ _n ; S) is a circulant and, by Lemma 11.2.3, its eigenvalues are

$$\{{\omega }^{{s}_{1} } + {\omega }^{{s}_{2} } + \cdots + {\omega }^{{s}_{p} }\, :\, \omega = \text{ an}\ n\text{ th root of unity}\}.$$

11.9.2 Unitary Cayley Graphs

We now take S( ⊂ ℤ _n ) to be the set U _n of numbers less than n and prime to n. Note that (a, n) = 1 if and only if (n − a, n) = 1. The corresponding Cayley graph Cay(ℤ _n ; U _n ) is denoted by X _n and called the unitary Cayley graph mod n (Note that U _n is the set of multiplicative units in ℤ _n .) Now suppose that (n ₁, n ₂) = 1. What is the Cayley graph ${X}_{{n}_{1}{n}_{2}}?$ Given $x \in {\mathbb{Z}}_{{n}_{1}{n}_{2}}$, there exist unique $c \in {\mathbb{Z}}_{{n}_{1}}$ and $d \in {\mathbb{Z}}_{{n}_{2}}$ such that

$$x \equiv c(\mathrm{mod}\ {n}_{1})\quad \mathrm{and}\quad x \equiv d(\mathrm{mod}\ {n}_{2}).$$

(11.5)

Conversely, given $c \in {\mathbb{Z}}_{{n}_{1}}$ and $d \in {\mathbb{Z}}_{{n}_{2}}$, by the Chinese Remainder Theorem [5], there exists a unique $x \in {\mathbb{Z}}_{{n}_{1}{n}_{2}}$ satisfying (11.5). Define $f : {\mathbb{Z}}_{{n}_{1}{n}_{2}} \rightarrow {\mathbb{Z}}_{{n}_{1}} \times {\mathbb{Z}}_{{n}_{2}}$ (the direct product of the groups ${Z}_{{n}_{1}}$ and ${Z}_{{n}_{2}}$) by setting f(x) = (c, d). Then f is an additive group isomorphism with $f({U}_{{n}_{1}{n}_{2}}) = {U}_{{n}_{1}} \times {U}_{{n}_{2}}$.

Two vertices a and b are adjacent in ${\mathbb{Z}}_{{n}_{1}{n}_{2}}$ if and only if $a - b \in {U}_{{n}_{1}{n}_{2}}$. Let

$$\begin{array}{rcl} a \equiv {a}_{1}(\mathrm{mod}\ {n}_{1}),& & b \equiv {b}_{1}(\mathrm{mod}\ {n}_{1}) \\ a \equiv {a}_{2}(\mathrm{mod}\ {n}_{2})& \quad \mathrm{and}\quad & b \equiv {b}_{2}(\mathrm{mod}\ {n}_{2})\end{array}.$$

Then (a − b, n ₁ n ₂) = 1 if and only if (a − b, n ₁) = 1 = (a − b, n ₂); equivalently, (a ₁ − b ₁, n ₁) = 1 = (a ₂ − b ₂, n ₂) or, in other words, ${a}_{1}{b}_{1} \in E({X}_{{n}_{1}})$ and ${a}_{2}{b}_{2} \in E({X}_{{n}_{2}})$. Thus, we have the following result.

Theorem 11.9.1.

If (n ₁ ,n ₂ ) = 1, the unitary Cayley graph ${X}_{{n}_{1}{n}_{2}}$ is isomorphic to ${X}_{{n}_{1}} \times {X}_{{n}_{2}}$ (where × on the right stands for direct product of graphs).

11.9.3 Spectrum of the Cayley Graph X _n

The adjacency matrix of X _n , as noted before, is a circulant, and hence the spectrum (λ₁, λ₂, …, λ _n ) of X _n (see the proof of Lemma 11.2.3) is given by

$${\lambda }_{r} ={ \sum \nolimits }_{\begin{array}{c}1\leq j\leq n, \\ (j,n)=1\end{array}}{\omega }^{rj},\quad \mathrm{where}\ \omega = {e}^{\frac{2\pi i} {n} }.$$

(11.6)

The sum in (11.6) is the well-known Ramanujan sum c(r, n) [123]. It is known that

$$c(r,n) = \mu ({t}_{r}) \frac{\phi (n)} {\phi ({t}_{r})},\ \mathrm{where}\ {t}_{r} = \frac{n} {(r,n)},\,0 \leq r \leq n - 1.$$

(11.7)

In relation (11.7), μ stands for the Möebius function. Further, as t _r divides n, ϕ(t _r ) divides ϕ(n), and therefore c(r, n) is an integer for each r. These remarks yield the following theorem.

Theorem 11.9.2.

The eigenvalues of the unitary Cayley graph X _n = Cay (ℤ _n ,U _n ) are all the integers c(r,n), 0 ≤ r ≤ n − 1.

For more spectral properties of Cayley graphs, the reader may consult Klotz and Sander [123].

11.10 Strongly Regular Graphs

Strongly regular graphs form an important class of regular graphs.

Definition 11.10.1.

A strongly regular graph with parameters (n, k, λ, μ) (for short: srg(n, k, λ, μ)) is a k-regular connected graph G of order n with the following properties:

• (i) Any two adjacent vertices of G have exactly λ common neighbors in G.

• (ii) Any two nonadjacent vertices of G have exactly μ common neighbors in G.

Example 11.10.2.

• (i) The cycle C ₅ is an srg(5, 2, 0, 1).

• (ii) The Petersen graph P is an srg(10, 3, 0, 1).

• (iii) The cycle C ₆ is a regular graph that is not strongly regular.

• (iv) The Clebesch graph is an srg(16, 5, 0, 2) (see Fig. 11.2).

• (v) Let q ≡ 1(mod 4) be an odd prime power. The Paley graphP(q) of order q is the graph with vertex set GF(q), the Galois field of order q, with two vertices adjacent in P(q) if and only if their difference is a nonzero square in GF(q). P(q) is an $srg(q, \frac{1} {2}(q - 1), \frac{1} {4}(q - 5), \frac{1} {4}(q - 1))$.

Our next theorem gives a necessary condition for the existence of a strongly regular graph with parameters (n, k, λ, μ).

Theorem 11.10.3.

If G is a strongly regular graph with parameters (n,k,λ,μ), then

$$k(k - \lambda - 1) = \mu (n - k - 1).$$

Proof.

We prove this theorem by counting the number of induced paths on three vertices in G having the same vertex v of G as an end vertex in two different ways. There are k neighbors w of v in G. For each such w, there are λ vertices that are common neighbors of v and w. Each of the remaining k − 1 − λ neighbors of w induces a P ₃ with v as an end vertex. As this is true for each neighbor w of v in G, there are k(k − λ − 1) paths of length 2 with v as an end vertex.

Fig. 11.2

Clebsch Graph

We now compute this number in a different way. There are n − 1 − k vertices x of G that are nonadjacent to v. Each such pair v, x is commonly adjacent to μ vertices of G. Each one of these μ vertices gives rise to an induced P ₃ in G with v as an end vertex. This number is μ(n − k − 1). □

Suppose now λ = k − 1. As G is k-regular, ab ∈ E(G), implies that every vertex c adjacent to a is adjacent to b, and every vertex d adjacent to b is adjacent to a. Now consider the adjacent pair b, c. Then c must be adjacent to d and so on. Thus, each component of G must be a clique (complete subgraph) of size k.

A similar reasoning applies when μ = 0. Further, if k = n − 1, G is K _n , the clique on n vertices. We treat these cases (that is, the cases when λ = k − 1; μ = 0; k = n − 1) as degenerate cases.

Theorem 11.10.4.

If G is an srg(n,k,λ,μ), then its complement G^c of G is an srg(n,n − 1 − k,n − 2 − 2k + μ,n − 2k + λ).

Proof.

Trivially, G ^c is n − 1 − k regular. Suppose now uv ∈ E(G ^c). Then uv∉E(G). There are k vertices adjacent to u in G and k vertices adjacent to v in G. Out of these 2k vertices, μ are common vertices adjacent to both u and v in G ^c. Hence, there are (n − 2) − (2k − μ) = n − 2 − 2k + μ vertices commonly adjacent to u and v in G ^c. By a similar argument, if uv∉E(G ^c), u and v are commonly adjacent to n − 2k + λ vertices in G ^c. Thus, G ^c is an srg(n, n − 1 − k, n − 2 − 2k + μ, n − 2k + λ). □

We now present another necessary condition for a graph to be strongly regular.

Theorem 11.10.5.

If G is an srg(n,k,λ,μ), with μ > 0, then the two numbers

$$\frac{1} {2}\left (n - 1 \pm \frac{(n - 1)(\mu - \lambda ) - 2k} {\sqrt{{(\mu - \lambda )}^{2 } + 4(k - \mu )}}\right )$$

are both nonnegative integers.

Proof.

We prove the theorem by showing that the two numbers are (algebraic) multiplicities of eigenvalues of G. If A is the adjacency matrix of G, J − I − A (where J is the all-1 matrix of order n) is the adjacency matrix of G ^c. What is the (i, j)th entry of A ²? If i = j, the number of v _i -v _j walks of length 2 is k since they are all of the form v _i v _p v _i , and there are k adjacent vertices v _p to v _i in G. Hence, each diagonal entry of A ² is k.

Now let i≠j. Then v _i v _j  ∈ E(G) if and only if v _i v _j ∉E(G ^c). Hence if i≠j, the (i, j)th entry is 1 or 0 in A if and only if the (i, j)th entry is 0 or 1, respectively, in J − I − A. Now there are exactly λ vertices commonly adjacent to v _i and v _j in G, and hence there are λ paths of length 2 in G. Hence, the 1’s in A will be replaced by λ in A ², and this is given by the matrix λA. Finally, let i≠j and v _i v _j ∉E(G). The number of v _i -v _j walks of length 2 in G is μ. The (i, j)th entry 0 in A should now be replaced by μ. This can be done by replacing the (i, j)th entry 1 in (J − I − A) by μ. Thus,

$${A}^{2} = kI + \lambda A + \mu (J - I - A).$$

(11.8)

Hence, if 1 is the column vector (1, 1, …, 1)^T of length n,

$${A}^{2}\mathbf{1} = (kI + \lambda A + \mu (J - I - A))\mathbf{1},$$

and this gives (as A 1 = k 1)

$${k}^{2}\mathbf{1} = k\mathbf{1} + \lambda k\mathbf{1} + \mu (n - 1 - k)\mathbf{1}.$$

(11.9)

Note that (11.9) yields another proof for Theorem 11.10.3.

Now as G is connected, 1 is a unique eigenvector corresponding to the eigenvalue k of G. Let x be any eigenvector corresponding to an eigenvalue θ≠k of G so that A x = θx. Then x is orthogonal to 1. Taking the product of both sides of (11.8) with x, we get (as J x = 0)

$${\theta }^{2}\mathbf{x} = {A}^{2}\mathbf{x} = k\mathbf{x} + \lambda \theta \mathbf{x} + \mu (-\mathbf{x} - \theta \mathbf{x}),$$

and therefore (as x is a nonzero vector),

$${\theta }^{2} - (\lambda - \mu )\theta - (k - \mu ) = 0.$$

(11.10)

This quadratic equation has two real roots (being eigenvalues of G), say, r and s, which are given by

$$\frac{1} {2}{[}(\lambda - \mu ) \pm \sqrt{{(\lambda - \mu )}^{2 } + 4(k - \mu )}{]}.$$

(11.11)

Denote the multiplicities of r and s by a and b. Now

$$\begin{array}{ccc} k + ra + sb = 0\ \text{ (sum of the eigenvalues)}, & \\ \mathrm{and}\ 1 + a + b = n\ \mathrm{ (total number of eigenvalues)}.& \end{array}$$

Solution of these equations gives

$$\begin{array}{rcl} a& =& -\frac{k + s(n - 1)} {r - s} , \\ b& =& \frac{k + r(n - 1)} {r - s} .\end{array}$$

(11.12)

Substitution of the values of r and s as given in (11.11) into (11.12) yields the two numbers given in the statement of Theorem 11.10.5. □

11.11 Ramanujan Graphs

Ramanujan graphs constitute yet another family of regular graphs. In recent times, a great deal of interest has been shown in Ramanujan graphs by researchers in diverse fields—graph theory, number theory, and communication theory.

Definition 11.11.1.

A Ramanujan graph is a k-regular connected graph G, k ≥ 2, such that if λ is any eigenvalue of G with | λ | ≠k, then $\lambda \leq 2\sqrt{k - 1}$.

Example 11.11.2.

The following graphs are Ramanujan graphs.

• 1. The complete graphs K _n , n ≥ 3.

• 2. Cycles C _n .

• 3. K _n, n . Here k = n, and $Sp({K}_{n,n}) = \left (\begin{array}{ccc} n& 0 & - n\\ 1 &2n - 2 & 1\end{array} \right )$.

• 4. The Petersen graph.

The Möebius ladderM _h is the cubic graph obtained by joining the opposite vertices of the cycle C _2h . By Exercise 14.6, we have

$${\lambda }_{j} = 2\cos \frac{\pi j} {h} + {(-1)}^{j},\quad 0 \leq j \leq 2h - 1.$$

Take h = 2p and j = 4p − 2. Then ${\lambda }_{4p-2} = 2\cos \frac{\pi (4p-2)} {2p} + 1 = 2\cos \frac{\pi } {p} + 1 > 2\sqrt{k - 1}$ (when p becomes large) $= 2\sqrt{2}$ (as k = 3). Hence, not every regular graph is a Ramanujan graph.

11.11.1 Why Are Ramanujan Graphs Important?

Let G be a k-regular Ramanujan graph of order n, and A its adjacency matrix. As A is symmetric, ℝ ⁿ has an orthonormal basis {u ₁, …, u _n } of eigenvectors of A. Since A 1 = k 1, we can take ${u}_{1} = {( \frac{1} {\sqrt{n}},\ldots , \frac{1} {\sqrt{n}})}^{\mathrm{T}}$. Let Sp(G) = { λ₁ ≥ λ₂ ≥ … ≥ λ _n }, where λ₁ = k and Au _i  = λ _i u _i , 1 ≤ i ≤ n. We can write A =  ∑_{i = 1}^{n}λ _i u _i u _i ^T. (This is seen from the fact that the matrices on the two sides when postmultiplied by u _j , 1 ≤ j ≤ n both yield λ _j u _j . ) More generally, as λ _i ^p, 1 ≤ i ≤ n are the eigenvalues of A ^p, we have

$${A}^{p} ={ \sum }_{i=1}^{n}{\lambda }_{ i}^{p}{u}_{ i}{u}_{i}^{\mathrm{T}}.$$

(11.13)

Let u _i  = ((u _i )₁, (u _i )₂, …, (u _i ) _r , …, (u _i ) _n ). Then the (r, s)th entry of A ^p =  ∑_{i = 1}^{n}λ _i ^p(u _i ) _r (u _i ^T) _s  =  ∑_{i = 1}^{n}λ _i ^p(u _i ) _r (u _i ) _s   : =  X _rs . As A is a binary matrix, X _r, s is a nonnegative integer. Moreover, as G is k-regular and connected, λ₁ = k. Set $\lambda (G)\ :{=\ \max }_{\vert {\lambda }_{i}\vert \neq k}\vert {\lambda }_{i}\vert$. Then,

$$\begin{array}{rcl}{ X}_{rs}& =& {\sum }_{i=1}^{n}{\lambda }_{ i}^{p}{({u}_{ i})}_{r}{({u}_{i})}_{s} \\ & =& {\vert }{\sum }_{i=1}^{n}{\lambda }_{ i}^{p}{({u}_{ i})}_{r}{({u}_{i})}_{s}{\vert } \\ &\geq & {k}^{p}{({u}_{ 1})}_{r}{({u}_{1})}_{s} -{\vert }{\sum }_{i=2}^{n}{\lambda }_{ i}^{p}{({u}_{ i})}_{r}{({u}_{i})}_{s}{\vert } \\ & =& {k}^{p} \frac{1} {\sqrt{n}} \frac{1} {\sqrt{n}} -\vert {\Sigma }_{0}\vert ,\ \mathrm{where}\ {\Sigma }_{0} ={ \sum }_{i=2}^{n}{\lambda }_{ i}^{p}{({u}_{ i})}_{r}{({u}_{i})}_{s}.\end{array}$$

(11.14)

Assume that G is not bipartite, so that λ _n ≠ − k (see Exercise 14.22). Hence, the eigenvalues λ₂, …, λ _n all satisfy | λ _i  | ≤ λ(G), and therefore

$$\begin{array}{rcl} \vert {\Sigma }_{0}\vert & \leq & \lambda {(G)}^{p}{ \sum }_{i=2}^{n}\vert {({u}_{ i})}_{r}\vert \vert {({u}_{i})}_{s}\vert \\ &\leq & \lambda {(G)}^{p}{(}{\sum }_{i=2}^{n}\vert {({u}_{ i})}_{r}^{2}\vert {{)}}^{1/2}{(}{\sum }_{i=2}^{n}\vert {({u}_{ i})}_{s}^{2}\vert {{)}}^{1/2}\quad \text{ (by Cauchy\textendash Schwarz inequality)} \\ & =& \lambda {(G)}^{p}{[}1 - {({u}_{ 1})}_{r}^{2}{{]}}^{1/2}{[}1 - {({u}_{ 1})}_{s}^{2}{{]}}^{1/2}\ ({\rm as\ the}\ {{u}_{i}}^\prime{\rm s\ are\ unit\ vectors}) \\ & =& \lambda {(G)}^{p}{[}1 - \frac{1} {n}{{]}}^{1/2}{[}1 - \frac{1} {n}{{]}}^{1/2} \\ & =& \lambda {(G)}^{p}{[}1 - \frac{1} {n}{]}\end{array}.$$

Hence, the (r, s)th entry of A ^p is positive if $\frac{{k}^{p}} {n} > \lambda {(G)}^{p}(1 - \frac{1} {n})$, that is, if $\frac{{k}^{p}} {\lambda {(G)}^{p}} > n - 1$.

This gives, on taking logarithms, that if

$$p > \frac{\log (n - 1)} {\log {(}k/\lambda (G){)}},$$

then every entry of A ^p is positive. Now the diameter of G is the least positive integer p for which A ^p is positive (see Exercise 14.3). Hence, the diameter D of G satisfies the inequality

$$D \leq \frac{\log (n - 1)} {\log (k/\lambda (G))} + 1.$$

Thus, we have proved the following theorem.

Theorem 11.11.3 (Chung [37]). 

Let G be a k-regular connected nonbipartite graph with n vertices and diameter D. Then

$$D \leq \dfrac{\log (n - 1)} {\log {(}k/\lambda (G){)}} + 1.$$

(11.15)

Now assume that G is bipartite and k-regular. Then λ₁ = k and λ _n  =  − k so that | λ₁ |  = k =  | λ _n  | . Working as above, one gets the following inequality in this case:

$$D \leq \dfrac{\log (n - 2)/2} {\log {(}k/\lambda (G){)}} + 2.$$

(11.16)

The last two inequalities show that to minimize D, one has to minimize $\lambda (G) {=\max }_{\vert {\lambda }_{i}\vert \neq k}\vert {\lambda }_{i}\vert$. Such graphs are useful in communication theory—the smaller the diameter, better the communication. In the case of Ramanujan graphs, we demand that $\lambda (G) \leq 2\sqrt{k - 1}$. Hence, Ramanujan graphs with sufficiently small values for λ(G) could be used in cost-efficient communication networks.

We mention that in constructing k-regular Ramanujan graphs G for a fixed k ≥ 2, it is not possible to bring down the upper bound $2\sqrt{k - 1}$ for λ(G) since as per a result of Serre [132], for ε > 0, there exists a positive constant c = c(k, ε), which depends only on k and ε such that the adjacency matrix of every k-regular graph on n vertices has at least cn eigenvalues larger than $(2 - \epsilon )\sqrt{k - 1}$ (see also [137]). Again, not every unitary Cayley graph is a Ramanujan graph. Unitary Cayley graphs that are Ramanujan graphs have been completely characterized by Droll [58].

We conclude this section with a remark that explains the name “Ramanujan graph” (see also [162]). The question that one may ask is the following: Does there exist a sequence {G _i } of graphs with an increasing number of vertices, satisfying the bound $\lambda ({G}_{i}) \leq 2\sqrt{k - 1}?$ That is, can we give an explicit construction for such a sequence of Ramanujan graphs? The only case known for which such sequences have been constructed is when k − 1 equals a prime power. In all these cases, the proof that the eigenvalues satisfy the required bound is by means of the Ramanujan’s conjecture in the theory of modular forms, proved by Deligne [48] in 1974 in the case when k − 1 is a prime, and by the work of Drinfeld [57] in the case when k − 1 is a prime power. This explains how Ramanujan’s name has entered into the definition of these graphs.

We have only touched upon the periphery of Ramanujan graphs. A deeper study of Ramanujan graphs requires expertise in number theory. Interested readers can refer to the two survey articles by Ram Murty ([162, 163]) and the relevant references contained therein.

11.12 The Energy of a Graph

11.12.1 Introduction

In this section, we discuss another application of eigenvalues of graphs. “The energy of a graph” is a concept borrowed from chemistry. Every chemical molecule can be represented by means of its corresponding molecular graph: Each vertex of the graph corresponds to an atom of the molecule, and two vertices of the graph are adjacent if and only if there is a bond between the corresponding molecules (the number of bonds being immaterial). The π-electron energy of a conjugated hydrocarbon, as calculated with the “Hückel molecular orbital (HMO) method,” coincides with the energy (as we are going to define below) of the corresponding graph. Conjugated hydrocarbons are of great importance for science and technology. A conjugated hydrocarbon can be characterized as a molecule composed entirely of carbon and hydrogen atoms in which every carbon atom has exactly three neighbors (which are either carbon or hydrogen atoms). See Fig. 11.3, which gives the structure of butadiene.

Fig. 11.3

Butadiene

The graph corresponding to a conjugated hydrocarbon is taken as follows (which is somewhat different from our description of molecular graphs given earlier in Chap. 1, and there are valid theoretical reasons for doing this): Every carbon atom is represented by a vertex and every carbon–carbon bond by an edge. Hydrogen atoms are ignored. Thus, the graph for the molecule of Fig. 11.3 is the path P ₄ on four vertices. These graphs of the conjugated hydrocarbons are connected and their vertex degrees are at most 3. However, for our general definition of graph energy, there is no such restriction and our graphs are quite general—only, they are simple. The pioneer in this area is Ivan Gutman. Readers interested in doing research in graph energy should consult [82–84].

Definition 11.12.1.

The energy of a graph G is the sum of the absolute values of its eigenvalues.

Hence if λ₁, λ₂, …, λ _n are the eigenvalues of a graph G of order n, the energy $\mathcal{E}(G)$ of G is given by

$$\mathcal{E}(G) ={ \sum }_{i=1}^{n}\vert {\lambda }_{ i}\vert .$$

Example 11.12.2.

• 1. The energy of the complete graph ${K}_{n} = \mathcal{E}({K}_{n}) = (n - 1) + (n - 1)\vert - 1\vert = 2(n - 1)$.

• 2. The spectrum of the Petersen graph is $\left (\begin{array}{ccc} 3&1& - 2\\ 1 &5 & 4\end{array} \right )$. (See Exercise 14.14). Hence, its energy is 1. 3 + 5. 1 + 4. 2 = 16.

11.12.2 Maximum Energy of k-Regular Graphs

Theorem 11.12.3.

The maximum energy of a k-regular graph G of order n is $k + \sqrt{k(n - 1)(n - k)}$ .

Proof.

Let Sp(G) = { λ₁ = k, λ₂, …, λ _n }. Apply Cauchy–Schwarz’s inequality to the two vectors ( | λ₂ | , | λ₃ | , …, | λ _n  | ) and (1, 1, …, 1). This gives

$${ \left ({\sum }_{i=2}^{n}\vert {\lambda }_{ i}\vert .1\right )}^{2} \leq \left ({\sum }_{i=2}^{n}\vert {\lambda }_{ i}{\vert }^{2}\right )\left ({\sum }_{i=2}^{n}{1}^{2}\right )$$

so that (as ∑ _i = 1 ⁿλ _i ² = 2m)

$${(}\mathcal{E}(G) - {\lambda }_{1}{)} \leq \sqrt{(2m - {\lambda }_{1 }^{2 })(n - 1)}.$$

Therefore, $\mathcal{E}(G) \leq F({\lambda }_{1})$, where

$$F({\lambda }_{1}) := {\lambda }_{1} + \sqrt{(2m - {\lambda }_{1 }^{2 })(n - 1)}.$$

(11.17)

If G is k-regular, by Theorem 11.5.1, λ₁ = k, and hence

$$\begin{array}{rcl} \mathcal{E}(G)& \leq & k + \sqrt{(n - 1)(2m - {k}^{2 } )} \\ & =& k + \sqrt{k(n - 1)(n - k)}\quad (\mathrm{as}\ 2m = nk).\end{array}$$

(11.18)

□

The same method yields for a general (not necessarily regular) graph the following theorem of Gutman [84].

Theorem 11.12.4.

Let G be a graph of order n and size m. Then

$$\mathcal{E}(G) \leq \frac{2m} {n} + \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}}.$$

(11.19)

Proof.

As usual, let A stand for the adjacency matrix of G. We have

$$\sqrt{\frac{2m} {n}} \leq \frac{2m} {n} \leq {\lambda }_{1} \leq \sqrt{2m}.$$

(11.20)

Relation (11.20) is a consequence of the fact that 0 < λ₁ ² ≤ λ₁ ² + ⋯ + λ _n ² = 2m and Exercise 14.10. Moreover, by (11.17), $\mathcal{E}(G) \leq F({\lambda }_{1})$. Now the function $F(x) = x + \sqrt{(n - 1)(2m - {x}^{2 } )}$ is strictly decreasing in the interval $\sqrt{\frac{2m} {n}} < x \leq \sqrt{2m}$. Hence, by inequality (11.20),

$$\mathcal{E}(G) \leq F({\lambda }_{1}) \leq F{(}\frac{2m} {n} {)} = \frac{2m} {n} + \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}}.$$

(11.21)

□

Theorem 11.12.5.

If 2m ≥ n and G is a graph on n vertices and m edges, then $\mathcal{E}(G) = \frac{2m} {n} + \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}}$ if and only if G is either $\frac{n} {2} {K}_{2},\,{K}_{n}$ or a non-complete connected strongly regular graph for which the spectrum {λ ₁ > λ ₂ ≥… ≥ λ _n } has the property that $\vert {\lambda }_{2}\vert = \vert {\lambda }_{3}\vert = \ldots = \vert {\lambda }_{n}\vert = \sqrt{\frac{2m-{(} \frac{2m} {n}{ {)}}^{2}} {n-1}}$ .

Proof.

We have already seen in Theorem 11.12.4 that $\mathcal{E}(G) \leq F{(}\frac{2m} {n} {)} = \frac{2m} {n} + \sqrt{(n - 1){(} 2m - {(} \frac{2m} {n}{ {)}}^{2}{)}}$. If $G \simeq \frac{n} {2} {K}_{2}$, then $Sp(G) = \left (\begin{array}{cc} 1 & - 1\\ \frac{n} {2} & \frac{n} {2} \end{array} \right )$, and if G ≃ K _n , then $Sp({K}_{n}) = \left (\begin{array}{cc} 1& - 1\\ 1 &n - 1 \end{array} \right )$. In both cases, it is routine to check that $\mathcal{E}(G) = F(\frac{2m} {n} )$. If the third alternative holds, G is regular of degree (see Theorem 11.5.1) $\frac{2m} {n} = {\lambda }_{1}$, and hence $\mathcal{E}(G),$ which, by hypothesis, is equal to ${\lambda }_{1} + (n - 1)\sqrt{\frac{2m-{[} \frac{2m} {n}{ {]}}^{2}} {n-1}} = \frac{2m} {n} + \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}} = F{(}\frac{2m} {n} {)}$, and therefore equality holds in (11.19).

Conversely, assume that equality is attained in (11.19) for some graph G. Then (11.20) and (11.21) together with the strict decreasing nature of F(x) imply that λ₁ (maximum degree of G) $= \frac{2m} {n}$ (average degree of G). Hence, G must be regular of degree $\frac{2m} {n}$. Moreover, in this case, equality must be attained in the Cauchy–Schwarz inequality applied in the proof of Theorem 11.12.3 and hence ( | λ₂ | , …, | λ _n  | ) is a scalar multiple of (1, 1, …, 1). In other words, | λ₂ |  = … =  | λ _n  | . Consequently, $\vert {\lambda }_{2}\vert + \cdots + \vert {\lambda }_{n}\vert = (n - 1)\vert {\lambda }_{i}\vert = \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}}$, and therefore, $\vert {\lambda }_{i}\vert = \sqrt{\frac{{[} 2m-{(} \frac{2m} {n}{ {)}}^{2}{]}} {n-1}} ,\,2 \leq i \leq n$. This results in three cases.

Case (i). :

The graph G has two distinct eigenvalues λ ₁ and λ ₂ with |λ ₁ | = |λ ₂ |. [If all the eigenvalues of G are equal, they must all be zero (as ∑λ _i  = 0) and hence G must be the totally disconnected graph K _n ^c.] Let m ₁ and m ₂ be the multiplicities of λ₁ and λ₂, respectively. Then 0 = tr(A) = m ₁λ₁ + m ₂λ₂ and hence m ₁ = m ₂. This gives λ₂ =  − λ₁, and therefore ${\lambda }_{2} = -\sqrt{\frac{{[} 2m-{(} \frac{2m} {n}{ {)}}^{2}{]}} {n-1}} = -{\lambda }_{1} = -\frac{2m} {n} ,$ which reduces to n = 2m. If G has ω components, $\frac{n} {2} = m \geq n - \omega ,$ and hence $\omega \geq \frac{n} {2}$. As G has no isolated vertices, $\omega = \frac{n} {2} ,$ and hence $G \simeq \frac{n} {2} {K}_{2}$.

Case (ii). :

G has two eigenvalues with distinct absolute values. Then | λ₁ | ≠ | λ₂ | ( =  | λ₃ |  = … =  | λ _n  | ). Now tr(A) = 0 gives $\frac{2m} {n} + (n - 1)\sqrt{\frac{{[} 2m-{(} \frac{2m} {n}{ {)}}^{2}{]}} {n-1}} = 0,$ which gives, on simplification, $m = \frac{n(n-1)} {2}$. Hence, G ≃ K _n .

Case (iii). :

G has exactly three distinct eigenvalues with distinct absolute values equal to $\frac{2m} {n}$ and $\sqrt{\frac{{[} 2m-{(} \frac{2m} {n}{ {)}}^{2}{]}} {n-1}}$ .

The conclusion in this case is an immediate consequence of a result of Shrikhande and Bhagwandas [172], which states that a k-regular connected graph G of order n is strongly regular if and only if it has three distinct eigenvalues α₁ = k, α₂, α₃ and if G is an srg(n, k, λ, μ), then λ = k + α₂α₃ + α₂ + α₃, and μ = k + α₂α₃. Hence, G must be connected (as the multiplicity of k is 1) and strongly regular.□

Remark 11.12.6.

Theorems 11.12.4 and 11.12.5 deal with the case 2m ≥ n. If 2m < n, the graph G is disconnected with n − 2m isolated vertices. Removal of these isolated vertices from G results in a subgraph G′ of G with n′ = 2m vertices and hence G′ ≃ mK ₂ by Theorem 11.12.5. Therefore, in this case, G ≃ mK ₂ ∪K _n − 2m ^c.

We conclude this section with the theorem of Koolen and Moulton [124] on the characterization of maximum energy graphs with a given number n of vertices.

Theorem 11.12.7.

Let G be a graph on n vertices. Then $\mathcal{E}(G) \leq \frac{n} {2} (1 + \sqrt{n})$ with equality holding if and only if G is a strongly regular graph with parameters $(n, \frac{n+\sqrt{n}} {2} , \frac{n+2\sqrt{n}} {4} , \frac{n+2\sqrt{n}} {4} )$ .

Proof.

First, assume that 2m ≥ n. By Theorem 11.12.4,

$$\mathcal{E}(G) \leq F(m) = \frac{2m} {n} + \sqrt{(n - 1){[} 2m - {(} \frac{2m} {n}{ {)}}^{2}{]}}.$$

(11.22)

We are interested in finding the values of m for which F(m) attains the maximum. This is done by the methods of elementary calculus: F′(m) = 0 gives 16nm ² − 8n ³ m + (n ⁵ − n ⁴) = 0, and this is the case when $m = \frac{{n}^{2}\pm n\sqrt{n}} {4}$. Further, F′(m) < 0 only for $m = \frac{{n}^{2}+n\sqrt{n}} {4}$. Hence, f(m) attains its maximum when $m = \frac{{n}^{2}+n\sqrt{n}} {4}$, and for this value of m, $F(\frac{2m} {n} ) = F(\frac{n+\sqrt{n}} {2} ) = \frac{{n}^{3}+\sqrt{n}} {2}$.

Next, assume that G is a strongly regular graph with parameters given in Theorem 11.12.5. As G is k-regular with $k = \frac{n+\sqrt{n}} {2}$, by Theorem 11.12.3,

$$\begin{array}{rcl} \mathcal{E}(G)& =& k + \sqrt{k(n - 1)(n - k)} \\ & =& \frac{n + \sqrt{n}} {2} + \sqrt{\left (\frac{n + \sqrt{n} } {2} \right )(n - 1)\left (n -\frac{n + \sqrt{n}} {2} \right )} \\ & =& \frac{{n}^{3/2} + n} {2} ,\,\text{ the maximum possible energy.} \\ \end{array}$$

Finally, assume that G attains the maximum possible energy. We have to show that G is an $srg(n, \frac{n+\sqrt{n}} {2} , \frac{n+2\sqrt{n}} {4} , \frac{n+2\sqrt{n}} {4} )$. By Theorem 11.12.5, three cases arise:

Case (i). :

$\frac{n(1+\sqrt{n})} {2} = \mathcal{E}(G) = \mathcal{E}(\frac{n} {2} {K}_{2}) = n$. This case cannot arise (as n > 1).

Case (ii). :

$\frac{n(1+\sqrt{n})} {2} = \mathcal{E}(G) = \mathcal{E}({K}_{n}) = 2(n - 1)$. This gives (n − 1)(n − 4)² = 0 and hence G ≃ K ₄, and corresponds to a degenerate strongly regular graph.

Case (iii). :

G is a strongly regular graph with two eigenvalues λ₂ and λ₃ (distinct from ${\lambda }_{1} = \frac{2m} {n}$) both having the same absolute value $\sqrt{\frac{{[} 2m-{(} \frac{2m} {n}{ {)}}^{2}{]}} {n-1}}$.

Recall that λ₂ and λ₃ are the roots of the quadratic equation (see (11.10) of Theorem 11.10.5) x ² + (μ − λ)x + (μ − k) = 0, where $k = {\lambda }_{1} = \frac{2m} {n}$. Now λ₂ + λ₃ = 0 gives λ = μ. Moreover, λ₂λ₃ = μ − k gives $\mu -\frac{2m} {n} = {\lambda }_{2}{\lambda }_{3} = -{(}\frac{2m-{(\frac{2m} {n} )}^{2}} {n-1} {)}$. Substituting $m = \frac{nk} {2} = n\frac{(n+\sqrt{n})} {2}$ and simplifying, we get $\mu = \frac{n+2\sqrt{n}} {4} = \lambda$. Hence, G is strongly regular with parameters $(n, \frac{n+\sqrt{n}} {2} , \frac{n+2\sqrt{n}} {4} , \frac{n+2\sqrt{n}} {4} )$. □

Finally, if 2m < n, by Remark 11.12.6, $\mathcal{E}(G) \leq 2m < n < \frac{n(1+\sqrt{n})} {2} ,\,n \geq 1$. Hence, the case $2m = \frac{n(1+\sqrt{n})} {2}$ cannot arise.

11.12.3 Hyperenergetic Graphs

The complete graph K _n , as seen in Example 11.12.2, has energy 2(n − 1). It was conjectured by Gutman [82] that if G is any graph of order n, then $\mathcal{E}(G) \leq 2(n - 1) = \mathcal{E}({K}_{n})$. This was disproved by many: Gutman et al. in [86] and Walikar et al. in [189]. Graphs G of order n for which $\mathcal{E}(G) > 2(n - 1)$ have been called hyperenergetic graphs.

Example 11.12.8.

• (i) K _n is nonhyperenergetic.

• (ii) All cubic graphs are nonhyperenergetic. If G is cubic, k = 3. Hence, by Theorem 11.12.3, $\mathcal{E}(G) \leq 3 + \sqrt{3(n - 1)(n - 3)}$. Now, $3 + \sqrt{3(n - 1)(n - 3)} \leq 2(n - 1)$ if and only if n ² − 8n + 16 ≥ 0, which is true as n ≥ 4 for any cubic graph.

• (iii) If n ≥ 5, L(K _n ) is hyperenergetic.

• (iv) If n ≥ 4, L(K _n, n ) is hyperenergetic.

We leave the proofs of (iii) and (iv) as exercises (See [188]).

Theorem 11.12.9 (Stevanović and Stanković [173]). 

The graph K _n − H, where H is a Hamilton cycle of K _n , n ≥ 10 is hyperenergetic.

Proof.

Let H = (v ₁ v ₂ …v _n ). Then the adjacency matrix A of K _n  − H is a circulant with first row (0 0 1 1 … 1 0). Hence, by Lemma 11.2.3,

$$\begin{array}{rcl} Sp({K}_{n} - H)& =& \{{\omega }^{2r} + {\omega }^{3r} + \cdots + {\omega }^{(n-2)r}\, :\, \omega = \mathrm{aprimitive}\ n\mathrm{throotofunity}\}. \\ & =& \left \{\begin{array}{l} n - 3,\ \mathrm{if}\ \omega = 1 \\ - 1 - {\omega }^{r} - {\omega }^{r(n-1)},\ \mathrm{if}\ \omega \neq 1,\,1 \leq r \leq n - 1. \end{array} \right . \\ & =& \left \{\begin{array}{l} n - 3,\ \mathrm{if}\ \omega = 1 \\ - 1 - 2\cos \frac{2\pi r} {n} ,\ \mathrm{if}\ \omega \neq 1,\,1 \leq r \leq n - 1. \end{array} \right. \end{array}.$$

Hence, $\mathcal{E}({K}_{n} - H) = (n - 3) +{ \sum }_{r=1}^{n-1}{\vert }- 1 - 2\cos \frac{2\pi r} {n} {\vert },$ which shows that (by the definition of Riemann integral) $\mathcal{E}({K}_{n} - H) \rightarrow (n - 3) + \frac{n-1} {2\pi } { \int \nolimits }_{0}^{2\pi }{\vert }- 1 - 2\cos x{\vert }\mathrm{d}x$ as n → ∞.

Hence,

$$ \begin{array}{rcl} {\lim }_{n\rightarrow \infty }\frac{\mathcal{E}({K}_{n} - H)} {n - 1} & =& 1 + \frac{1} {2\pi }{\int \nolimits }_{0}^{2\pi }{\vert }- 1 - 2\cos x{\vert }\mathrm{d}x \\ & =& \frac{4\sqrt{3}} {2\pi } + \frac{1} {3} \\ & \approx & 2.43599, \\ \end{array}$$

which shows that there exists n ₀ such that for $n \geq {n}_{0},\,\mathcal{E}({K}_{n} - H) > 2(n - 1)$. Computations show that n ₀ = 10. □

Theorem 11.12.9 disproves a conjecture given in [10].

Graph $\overline{C}(n;{k}_{1},\ldots ,{k}_{r}),\,{k}_{1} < {k}_{2} < \ldots < {k}_{r} < \frac{n} {2} ,\,{k}_{i} \in \mathbb{N}$ for each i is the circulant graph with vertex set V = { 0, 1, …, n − 1} such that a vertex u is adjacent to all the vertices of V ∖ {u}, except u ± k _i (mod n), i = 1, 2, …, n. Note that K _n  − H is just $\overline{C}(n;1)$. Stevanović and Stanković also generalized Theorem 11.12.9 as follows:

Theorem 11.12.10 (Stevanović and Stanković [173]). 

Given k ₁ < k ₂ < … < k _r , there exists n ₀ ∈ ℕ such that for n ≥ n ₀ , the graph $\overline{C}(n;{k}_{1},\ldots ,{k}_{r})$ is hyperenergetic.

11.12.4 Energy of Cayley Graphs

We now compute the energy of the unitary Cayley graph ${X}_{{p}^{\alpha }}$, where p is a prime. This can be done in two ways: In [10], Balakrishnan computed it using cyclotomic polynomials, while in [161], Ramaswamy and Veena computed it using Ramanujan sums. We present the latter method.

Theorem 11.12.11.

If p is a prime and $r \geq 1,\,\mathcal{E}({X}_{{p}^{\alpha }}) = 2\phi ({p}^{\alpha })$ .

Proof.

If α = 1, ${X}_{{p}^{\alpha }} = {X}^{p} = {K}_{p}$, and $\mathcal{E}({K}_{p}) = 2(p - 1) = 2\phi (p)$. So assume that α ≥ 2. By Theorem 11.9.2, $Sp({X}_{{p}^{\alpha }}) =\{ {\lambda }_{0},\ldots ,{\lambda }_{n-1}\},$ where

$${\lambda }_{r+1} = c(r,{p}^{\alpha }) = \mu ({t}_{ r})\frac{\phi ({p}^{\alpha })} {\phi ({t}_{r})} ,\ \mathrm{and}\ {t}_{r} = \frac{{p}^{\alpha }} {(r,{p}^{\alpha })},\,0 \leq r < {p}^{\alpha }.$$

Here μ is the Möebius function and ϕ is Euler’s totient function.

We consider three cases:

• (i) (r, p ^α) = p ^α. Then as r < p ^α, r = 0, t _r  = 1, and μ(t _r ) = 1, and therefore${\lambda }_{1} = \frac{\phi ({p}^{\alpha })} {\phi (1)} = {p}^{\alpha } - {p}^{\alpha -1}$.

• (ii) (r, p ^α) = 1. Then t _r  = p ^α and therefore μ(t _r ) = 0. Hence, λ _r + 1 = 0 in this case.

• (iii) If 1 < (r, p ^α) < p ^α, then (r, p ^α) = p ^m, 1 ≤ m ≤ α − 1. If (r, p ^α) = p ^α − 1, then r ∈ { 1. p ^α − 1, 2. p ^α − 1, …, (p − 1). p ^α − 1}, and hence t _r  = p and therefore μ(t _r ) =  − 1. Hence, ${\lambda }_{r+1} = -\frac{\phi ({p}^{\alpha })} {\phi (p)} = -{p}^{\alpha -1}$.

In all the other cases, μ(t _r ) = 0, and therefore λ _r + 1 = 0.

Hence, $Sp({X}_{{p}^{\alpha }}) = \left (\begin{array}{ccc} {p}^{\alpha } - {p}^{\alpha -1} & - {p}^{\alpha -1} & 0 \\ 1 & p - 1 &{p}^{\alpha } - p \end{array} \right )$, and therefore

$$\begin{array}{rcl} \mathcal{E}({X}_{{p}^{\alpha }})\,& =\, ({p}^{\alpha } - {p}^{\alpha -1}) + (p - 1){p}^{\alpha -1}& \\ & = 2({p}^{\alpha } - {p}^{\alpha -1}) & \\ & = 2\phi ({p}^{\alpha }). & \\ \end{array}$$

□

In [10], Theorem 11.12.11 has been applied to prove the next result, which in essence constructs for each n, a k-regular graph of order n for a suitable k whose energy is small compared to the maximum possible energy among the class of k-regular graphs.

Theorem 11.12.12 ([10]). 

For each ε > 0, there exist infinitely many n for each of which there exists a k-regular graph G of order n with k < n − 1 and $\frac{\mathcal{E}(G)} {B} < \epsilon$ , where B is the maximum energy that a k-regular graph on n vertices can attain.

Proof.

The Cayley graph ${X}_{{p}^{\alpha }}$ is ϕ(p ^α)-regular. The maximum energy that a k-regular graph can attain is (by Theorem 11.12.3) $B = k + \sqrt{k(n - 1)(n - k)}$. Taking n = p ^α and k = ϕ(p ^α), this bound becomes

$$\begin{array}{rcl} B& =& \phi ({p}^{\alpha }) + \sqrt{\phi ({p}^{\alpha } )({p}^{\alpha } - 1)({p}^{\alpha } - \phi ({p}^{\alpha } ))} \\ & =& ({p}^{\alpha } - {p}^{\alpha -1}) + \sqrt{({p}^{\alpha } - {p}^{\alpha -1 } )({p}^{\alpha } - 1){p}^{\alpha -1}} \\ & =& ({p}^{\alpha } - {p}^{\alpha -1}) + {p}^{\alpha -1}\sqrt{(p - 1)({p}^{\alpha } - 1})\end{array}.$$

Hence, $\frac{\mathcal{E}({X}_{{p}^{\alpha }})} {B} = \dfrac{2({p}^{\alpha } - {p}^{\alpha -1})} {({p}^{\alpha } - {p}^{\alpha -1}) + {p}^{\alpha -1}\sqrt{(p - 1)({p}^{\alpha } - 1)}} \,\,\,\,\, = \dfrac{2} {1 + \sqrt{1 + p + \cdots + {p}^{\alpha -1}}}\,\rightarrow 0$ as either p → ∞ or α → ∞. □

In [10], the following dual question was also raised: Given a positive integer n ≥ 3, and ε > 0, does there exist a k-regular graph G of order n such that $\frac{\mathcal{E}(G)} {B} > 1 - \epsilon$ for some k < (n − 1)? An affirmative answer to this question is given in [133] and [190], not for general n, but when n is a prime power p ^m ≡ 1(mod 4). Interestingly, both papers give the same example, namely, the Paley graph [see Example 11.10.2(v)].

Lemma 11.12.13.

$\mathcal{E}({G}_{1} \times {G}_{2}) = \mathcal{E}({G}_{1})\mathcal{E}({G}_{2})$ .

Proof.

From Corollary 11.8.7, it follows that if Sp(G ₁) = { λ₁, …, λ _p } and Sp(G ₂) = { μ₁, …, μ _q }, then Sp(G ₁ ×G ₂) = { λ _i μ _j  : 1 ≤ i ≤ p, 1 ≤ j ≤ q}. Hence,$\mathcal{E}({G}_{1}\times {G}_{2}) ={ \sum }_{i=1}^{p}{ \sum }_{j=1}^{q}{\vert }{\lambda }_{i}{\mu }_{j}{\vert } = {(}{\sum }_{i=1}^{p}{\vert }{\lambda }_{i}{\vert }{)}{(}{\sum }_{j=1}^{q}{\vert }{\mu }_{j}{\vert }{)} = \mathcal{E}({G}_{1}){\mathcal{E}}_{(}{G}_{2})$. □

Recall Theorem 11.9.2, which states that if $({n}_{1},{n}_{2}) = 1,\,{X}_{{n}_{1}{n}_{2}} \simeq {X}_{{n}_{1}} \times {X}_{{n}_{2}}$.

Corollary 11.12.14 ([161]). 

$\mathcal{E}({X}_{n}) = {2}^{k}\phi (n)$ , where k is the number of distinct prime factors of n.

Proof.

If p ₁ and p ₂ are distinct primes, then by Lemma 11.12.13, $\mathcal{E}({X}_{{p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}}}) = \mathcal{E}({X}_{{p}_{1}^{{\alpha }_{1}}})\mathcal{E}({X}_{{p}_{2}^{{\alpha }_{2}}}) = 2\phi ({p}_{1}^{{\alpha }_{1}}).2\phi ({p}_{2}^{{\alpha }_{2}}) = {2}^{2}\phi ({p}_{1}^{{\alpha }_{1}}{p}_{2}^{{\alpha }_{2}})$. By induction, it follows that if $n = {p}_{1}^{{\alpha }_{1}}\ldots {p}_{k}^{{\alpha }_{k}}$, where the p _i ’s are the distinct prime factors of n, $\mathcal{E}({X}_{n}) ={ \prod \nolimits }_{i=1}^{k}\mathcal{E}({p}_{i}^{{\alpha }_{i}}) ={ \prod \nolimits }_{i=1}^{k}2\phi ({p}_{i}^{{\alpha }_{i}}) = {2}^{k}\phi (n)$. □

The last result has also been proved in [108] using different techniques.

11.13 Energy of the Mycielskian of a Regular Graph

In this section we determine the energy of the Mycielskian of a k-regular graph G in terms of the energy of G. [We defined the Mycielskian μ(G) of a graph G in Chap. 7.] Some papers that deal with the energy of regular graphs are [4, 87, 109].

Theorem 11.13.1 ([13]). 

Let G be a k-regular graph on n vertices. Then the energy $\mathcal{E}(\mu (G))$ of μ(G) is given by

$$\mathcal{E}(\mu (G)) = \sqrt{5}\mathcal{E}(G) - (\sqrt{5} - 1)k + 2\vert {t}_{3}\vert ,$$

(11.23)

where $\mathcal{E}(G)$ is the energy of G and t ₃ is the unique negative root of the cubic

$${t}^{3} - k{t}^{2} - (n + {k}^{2})t + kn.$$

(11.24)

Proof.

Denote by A the adjacency matrix of G. As A is real symmetric, A = PDP ^T, where D is the diagonal matrix diag(λ₁, …, λ _n ), and P is an orthogonal matrix with orthonormal eigenvectors p _i ’s, that is, Ap _i  = λ _i p _i for each i. In particular, if e denotes the n-vector with all entries equal to 1, then ${p}_{1} = \frac{e} {\sqrt{n}}$ (G being regular). Hence, e ^T p _i =0 for i = 2, …, n, and consequently ${e}^{\mathrm{T}}P = [\sqrt{n},0,\ldots ,0]$.

Now, by the definition of μ(G), the adjacency matrix of μ(G) is the matrix of order (2n + 1) given by

$$\mu (A) = \left [\begin{array}{*{10}c} A& \,\,\,A &\,\,\,0\\ A & \,\,\,0 & \,\,\,e \\ 0 &\,\,\,0{e}^{\mathrm{T}} & \,\,\,0& \end{array} \right ].$$

Since A = PDP ^T, we have

$$\begin{array}{rcl} \mu (A)& = \left [\begin{array}{*{10}c} PD{P}^{\mathrm{T}} & \,\,\,PD{P}^{\mathrm{T}} & \,\,\,0 \\ PD{P}^{\mathrm{T}} & \,\,\,0 & \,\,\,e \\ 0 & {e}^{\mathrm{T}} & \,\,\,0\,\,\,& \end{array} \right ] & \\ & = \left [\begin{array}{*{10}c} P & \,\,\,0 &\,\,\,0\\ 0 &\,\,\,P &\,\,\,0 \\ 0 & \,\,\,0 &\,\,\,1 \end{array} \right ]\left [\begin{array}{*{10}c} D& \,\,\,D & \,\,\,0 \\ D& \,\,\,0 &\,\,\,{P}^{\mathrm{T}}e \\ 0 &\,\,\,{e}^{\mathrm{T}}P & \,\,\,0 & \end{array} \right ]\left [\begin{array}{*{10}c} {P}^{\mathrm{T}} & \,\,\,0 &\,\,\,0 \\ 0 &\,\,\,{P}^{\mathrm{T}} & \,\,\,0\\ 0 & \,\,\,0 &\,\,\,1 \end{array} \right ].& \\ \end{array}$$

As P is an orthogonal matrix, the spectrum of μ(A) is the same as the spectrum of

$$\begin{array}{rcl} & B = \left [\begin{array}{*{10}c} D& \,\,\,D & \,\,\,0 \\ D& \,\,\,0 &\,\,\,{[\sqrt{n},0,\ldots ,0]}^{\mathrm{T}} \\ 0 &\,\,\,[\sqrt{n},0,\ldots ,0]& \,\,\,0 & \end{array} \right ].& \\ \end{array}$$

The determinant of the characteristic matrix of B is given by

We now expand det(xI − B) along its first, (n + 1)th, and (2n + 1)th columns by Laplace’s method [61]. [Recall that Laplace’s method of expansion of a determinant T along any chosen set $\{{C}_{{j}_{1}},\ldots ,{C}_{{j}_{k}}\}$ of k columns (or rows) is a natural generalization of the usual expansion of a determinant along any column (or row). This expansion of T is given by ${\sum }_{({i}_{1},\ldots ,{i}_{k})}{(-1)}^{({i}_{1}+\cdots +{i}_{k})+({j}_{1}+\cdots +{j}_{k})}{U}_{k}{V }_{2n+1-k},$ where U _k is the determinant minor of T of order k common to the k rows ${R}_{{i}_{1}},\ldots ,{R}_{{i}_{k}}$, and the k columns ${C}_{{j}_{1}},\ldots ,{C}_{{j}_{k}}$, and V _{2n + 1 − k} is the complementary determinant minor of order (2n + 1 − k) obtained by deleting these sets of k rows and k columns from T.] In this expansion, only the 3 ×3 minor, which is common to the above three columns, and the 1-st, (n + 1)th, and (2n + 1)th rows are nonzero. This minor is

$${M}_{1} = \left \vert \begin{array}{ccc} x - {\lambda }_{1} & \,\,\, - {\lambda }_{1} & \,\,\,0 \\ - {\lambda }_{1} & \,\,\,0 &\,\,\, -\sqrt{n} \\ 0 &\,\,\, -\sqrt{n}& \,\,\,0\\ \end{array} \right \vert .$$

We now expand the complementary minor of M ₁ along the 2nd and (n + 2)th columns, and then the resulting complementary minor by the 3rd and (n + 3)th columns, and so on. These expansions give the spectrum of μ(A) to be the union of the spectra of the matrices

$$\left [\begin{array}{*{10}c} {\lambda }_{1} & \,\,\,{\lambda }_{1} & \,\,\,0 \\ {\lambda }_{1} & \,\,\,0 &\,\,\,\sqrt{n} \\ 0 &\,\,\,\sqrt{n}& \,\,\,0 \end{array} \right ],\,\left [\begin{array}{*{10}c} {\lambda }_{2} & \,\,\,{\lambda }_{2} \\ {\lambda }_{2} & \,\,\,0 \end{array} \right ],\,\ldots ,\,\left [\begin{array}{*{10}c} {\lambda }_{n}&\,\,\,{\lambda }_{n} \\ {\lambda }_{n}& \,\,\,0 \end{array} \right ].$$

(Another way to see this is to observe that B is orthogonally similar to the direct sum of the above n matrices.) Thus, the spectrum of μ(A) is

$$\left \{{t}_{1},\,{t}_{2},\,{t}_{3},\,{\lambda }_{2}\left (\dfrac{1 \pm \sqrt{5}} {2} \right ),\,\ldots ,\,{\lambda }_{n}\left (\dfrac{1 \pm \sqrt{5}} {2} \right )\right \},$$

where t ₁, t ₂, t ₃ are the roots of the cubic polynomial t ³ − kt ² − (n + k ²)t + kn (note that as G is k-regular, λ₁ = k), which has two positive roots and one negative root, say, t ₁ > t ₂ > 0 > t ₃.

Let $\mathcal{E}(G)$ denote the energy of G. Then $\mathcal{E}(G) ={ \sum \nolimits }_{i}\vert {\lambda }_{i}\vert = k + \vert {\lambda }_{2}\vert + \cdots + \vert {\lambda }_{n}\vert$. Hence, the energy of the Mycielskian μ(G) of G, when G is k-regular, is

$$\begin{array}{rcl} \mathcal{E}(\mu (G))& = \vert {t}_{1}\vert + \vert {t}_{2}\vert + \vert {t}_{3}\vert + {(}\vert \frac{1+\sqrt{5}} {2} \vert + \vert \frac{1-\sqrt{5}} {2} \vert {)}{(}\vert {\lambda }_{2}\vert + \cdots + \vert {\lambda }_{n}\vert {)}& \\ & = {t}_{1} + {t}_{2} + \vert {t}_{3}\vert + \sqrt{5}(\vert {\lambda }_{2}\vert + \cdots + \vert {\lambda }_{n}\vert ) & \\ & = k - {t}_{3} + \vert {t}_{3}\vert + \sqrt{5}(\mathcal{E}(G) - k) & \\ & = \sqrt{5}\mathcal{E}(G) - (\sqrt{5} - 1)k + 2\vert {t}_{3}\vert . & \\ \end{array}$$

□

11.13.1 An Application of Theorem 11.13.1

We have shown in Theorem 11.12.7 that the maximum energy that a graph G of order n can have is $\frac{{n}^{\frac{3} {2} }+n} {2}$ and that G has maximum energy if and only if it is a strongly regular graph with parameters ${(}n, \frac{n+\sqrt{n}} {2} , \frac{n+2\sqrt{n}} {4} , \frac{n+2\sqrt{n}} {4} {)}$. If n > 25 and G has maximum energy, $\mathcal{E}(G) > 3n > 2(n - 1)$ and hence G is hyperenergetic. Also, from (11.23), $\mathcal{E}(\mu (G)) > 3n\sqrt{(}5) - (\sqrt{(}5) - 1)k$ $+2\vert {t}_{3}\vert > 3n\sqrt{(}5) - (\sqrt{(}5) - 1)(n - 1)$ $> 2\sqrt{(}5)n + \sqrt{(}5) + 1 > 4n = 2[(2n + 1) - 1]$. Hence, the Mycielskians of maximal energy graphs of order n > 25 are all hyperenergetic. More generally, if G is a regular graph of order n and $\mathcal{E}(G) > 3n$, then μ(G) is hyperenergetic.

Example 11.13.2 ([13]). 

We now present two examples. Consider two familiar regular graphs, namely, the Petersen graph P and a unitary Cayley graph.

• 1. For P, n = 10, m = 15, and k = 3. The spectrum of P is $\left (\begin{array}{ccc} 3&1& - 2\\ 1 &5 & 4\end{array} \right )$. Hence, $\mathcal{E}(P) = 16$. Now for P, the polynomial (3.2) is t ³ − 3t ² − 19t + 30, and its unique negative root t ₃ is ≈ − 3. 8829. Hence, from (11.23), $\mathcal{E}(\mu (P)) \approx 16\sqrt{5} - (\sqrt{5} - 1)3 + 2(3.8829) = 39.8347 < 40 = 2(21 - 1)$, where 21 is the order of μ(P). Hence, μ(P) is non-hyperenergetic.

• 2. Consider the unitary Cayley graph $G = \mathrm{Cay}({\mathbb{Z}}_{210},U)$, U being the group of multiplicative units of the additive group $({\mathbb{Z}}_{210},+)$. Then $\mathcal{E}(G) = {2}^{4}\phi (210) = 768 > 3 \times 210 = 3n$. Here n = 210, k = ϕ(210) = 48. Hence, the polynomial (11.24) becomes t ³ − 48t ² − 2514t + 10, 080, for which the unique negative root t ₃ is ≈ − 34. 18. Hence, from (11.23), $\mathcal{E}(\mu (G)) \approx 768\sqrt{5}-(\sqrt{5}-1)48+2(34.18) = 1,726.352 > 2((2n+1)-1) = 840$. Thus, $\mu (\mathrm{Cay}({\mathbb{Z}}_{210},U))$ is hyperenergetic, as expected.

11.14 Exercises

• 14.1. Prove: For any nontrivial graph of order n with spectrum = { λ₁ ≥ λ₂ ≥ … ≥ λ _n }, λ _n  ≤ 0.

• 14.2. If G is k-regular, show that the number of connected components of G is the multiplicity of k as an eigenvalue of G.

• 14.3. Prove by induction on l that the number of v _i -v _j walks of length l in the connected graph G with V (G) = { v ₁, …, v _n } is the (i, j)th entry of A ^l. Hence show that the diameter of a connected graph G is the least positive integer d with A ^d > 0 (that is, all the entries of A ^d are positive).

• 14.4. Let G be a complete r-partite graph of order n with all parts of the same size. Find Sp(G). In particular, find Sp(K _p, p ), and compare it with Theorem 11.6.1.

• 14.5. If G and H are disjoint graphs, prove that Sp(G ∪H) = Sp(G) ∪Sp(H), and that ϕ(G ∪H; λ) = ϕ(G; λ)ϕ(H; λ).

• 14.6. The Möbius ladder M _h is the cubic graph obtained by joining the opposite vertices of the cycle C _2h . Show that the spectrum of the Möbius ladder is given by ${\lambda }_{j} = 2\cos \frac{\pi j} {h} + {(-1)}^{j},\,0 \leq j \leq 2h - 1.$

• 14.7. Find the spectrum of

  • (i) C ₄ ×C ₃,

  • (ii) K ₄ □K ₃,

  • (iii) L(K _4, 3) using Sachs’ theorem and hence comment on the results in (ii) and (iii).

• 14.8. If G is a bipartite graph of odd order, prove that detA is zero.

• 14.9. If G is bipartite and A is nonsingular, show that G has a perfect matching.

• 14.10. Let G be a graph of order n with adjacency matrix A. Prove that for any nonzero vector ${ x} \in {\mathbb{R}}^{n},\,{\lambda }_{1} = {\lambda }_{\max }(G) \geq \frac{(A \text{ x}, \text{ x})} {( \text{ x}, \text{ x})} \geq {\lambda }_{\min }(G) = {\lambda }_{n}$. In particular, show that ${\lambda }_{1} \geq \frac{2m} {n}$. (Hint: Use the fact that ℝ ⁿ has an orthonormal basis of eigenvectors of A.)

• 14.11. Find the spectrum of K ₄ ⊠ K ₃.

• 14.12. Show that a graph G on n vertices is connected if and only if each entry of (A + I)^n − 1 is positive.

• 14.13. Let λ₁, …, λ _r be the distinct eigenvalues of A, the adjacency matrix of a graph G of order n. Prove that the minimal polynomial of A is (x − λ₁)…(x − λ _r ). (Hint: Same as for Exercise 14.10.)

• 14.14. Using the fact that the Petersen graph P ≃ (L(K ₅))^c, show that $Sp(P) = \left (\begin{array}{ccc} 3&1& - 2\\ 1 &5 & 4\end{array} \right )$.

• 14.15. Is the Petersen graph a Cayley graph?

• 14.16. When n is a prime, show that the unitary Cayley graph X _n is the complete graph K _n .

• 14.17. Let G be a finite abelian group of order n. Prove that Sp(Cay(G; S)) = { λ_χ =  ∑_{s ∈ S}χ(s) : χ =  anirreduciblecharacterof G}. [Hint: Show that if G = { g ₁, …, g _n }, then (χ(g ₁), …, χ(g _n ))^T is an eigenvector of the adjacency matrix of the Cayley graph.]

• 14.18. If λ is any eigenvalue of a graph G of order n and size m, prove that $\lambda \leq \sqrt{\frac{2m} {n} (n - 1)}$. (Hint: Imitate the relevant steps given in the proof of Theorem 11.12.3.)

• 14.19. Show that the Petersen graph is a Ramanujan graph.

• 14.20. Show that the graphs K _n ×K _n ×… ×K _n p times, where p  ≥ 4, n  ≥ 3, form a family of regular graphs none of which is a Ramanujan graph.

• 14.21. Prove that if G is bipartite, then χ(G; x) is of the form ϕ(x ²) or xϕ(x ²). Hence, prove that if λ is an eigenvalue of a bipartite graph with multiplicity m(λ), then so is − λ.

• 14.22. Show that a connected k-regular graph is bipartite if and only if − k ∈ Sp(G). [Hint: Suppose − k ∈ Sp(G) and that (x ₁ = 1, x ₂, …, x _n )^T is an eigenvector corresponding to − k. Then if ${v}_{{i}_{1}},\ldots ,{v}_{{i}_{k}}$ are the neighbors of ${v}_{1},\,{x}_{{i}_{1}} = {x}_{{i}_{2}} = \ldots = {x}_{{i}_{k}} = -1$. If v _j is a neighbor of ${v}_{{i}_{1}},\,{x}_{j} = 1,$ etc.]

• 14.23. Establish the inequality (11.16).

• 14.24. Prove: If n ≥ 5, L(K _n ) is hyperenergetic.

• 14.25. Prove: If n ≥ 4, L(K _n, n ) is hyperenergetic.

11.15 Notes

Standard references for algebraic graph theory are [22, 47, 75].

Many variations of graph energy have been studied recently. Two types of energy of digraphs have been considered—skew energy of digraphs by taking the adjacency matrix of the digraph to be skew-symmetric [1, 171] and another by taking it to be the nonsymmetric (0, 1)-matrix [157]. The Laplacian matrix of a simple graph G is the matrix L = Δ − A, where Δ is the diagonal matrix whose diagonal entries are the degrees of the vertices of G, and A is the adjacency matrix of G. The energy of the matrix L is the Laplacian energy of G [85]. Yet another graph energy that has been studied is the distance energy of a graph [110]. It is the energy of the distance matrix D = (d _ij ) of G, where d _ij is the distance between the vertices v _i and v _j of G. More generally, Nikiforov has introduced and studied [147] the concept of the energy of any p by q matrix.