Relating Hypergraph Parameters of Generalized Power Graphs

Abstract

Graph parameters like the chromatic number, independence number, clique number and many others alongside with their corresponding adjacency matrix have been broadly studied and extended to hypergraphs classes. A generalized power graph $G^k_s$ of a graph G is a k-uniform hypergraph constructed by blowing up each vertex of G into a s-set of vertices and then adding k − 2s vertices of degree one to each edge, where k ≥ 2s. A natural question is whether there exists any relation between structural parameters and spectral parameters of $G^k_s$ with the respective parameters of the original graph G. In this paper we positively answer this question and investigate the parameters behavior.

1 Introduction

A hypergraph H = (V, E) is given by a vertex set V  and a set $E= \left \{ e:e \subseteq V \right \}$, whose elements are called (hyper) edges. A graph G = (V, E) is a hypergraph such that |e|≤ 2 for every e ∈ E.

Different aspects of a graph like clique number, vertex or edge coloring, matching, connectivity, have been widely studied in many areas and can be generalized to hypergraph theory, for example hypergraph coloring and strong hypergraph coloring, weak and strong vertex connectivity [4, 9]. In [1], the authors stated that strong hypergraph coloring captures many previously studied graph coloring properties. These different ways of expanding a graph parameter have attracted the attention of researchers: [9] studied the difference between weak and strong vertex connectivity; and [2, 7, 11] exclusively focused their work on a single parameter.

Spectral graph theory is another area that can be extended to hypergraphs. The goal of spectral graph theory is to study eigenvalues and eigenvectors of matrices associated with graphs finding information of structural properties of these graphs. Many graph matrices are studied in spectral graph theory which can also be extended to hypergraphs in different ways. The study of hypergraph matrices started in the 1990s with a generalization of the graph adjacency matrix [10], and new matrices are still being defined. In 2019, [3], a similar but different adjacency matrix of a hypergraph is defined, allowing a generalization of important spectral graph theory results to hypergraphs. Another approach to spectral hypergraph theory was given in 2012, [8], when it is proposed the study of hypergraphs through tensors.

This work aims to investigate the relation between hypergraph structural parameters and spectral parameters of a class of uniform hypergraphs, called generalized power graph, that was first considered in [15]. Recently, this class was studied by considering its tensor spectra [13,14,15].

Since these hypergraphs are constructed from a base graph, we discuss four main topics: the relation between hypergraph parameters with their respective graph parameters; the behavior of distinct variations of generalized graph parameters on this hypergraph class; the relation between the adjacency matrix of this hypergraph with matrices of the base graph; and new relations of hypergraph parameters and the adjacency matrix eigenvalues.

2 Preliminaries

A hypergraph H = (V, E) is k-uniform if |e| = k for every edge e ∈ E(H). A simple graph G = (V, E) is a 2-uniform hypergraph. In this work we consider only simple hypergraphs, i.e. it contains no loops (edges with |e| = 1) and no repeated edges. A null hypergraph contains no vertices (or no edges) and a hypergraph with only one vertex is called trivial. Two vertices in a hypergraph are adjacent if there is an edge which contains both vertices, and the degree of a vertex v ∈ V  is $d(v)=|\left \{ e: v \in e \right \}|$, the number of edges that contain v.

A path P in a hypergraph H is a vertex-edge alternating sequence: P = v ₀, e ₁, v ₁, e ₂, …, v _r−1, e _r, v _r such that v ₀, v ₁, …, v _r are distinct vertices; e ₁, e ₂, …, e _r are distinct edges; and v _i−1, v _i ∈ e _i, i = 1, 2, …, r. The length of a path P is the number of distinct edges. A hypergraph is connected if for any pair of vertices, there is a path which connects these vertices; it is not connected otherwise.

Let G be a graph and s ≥ 1 an integer. The s-extension G _s of G is a 2s-uniform hypergraph obtained from G by replacing each vertex v _i ∈ V  by a set $S_{v_i}= \left \{ v_{i1},\ldots ,v_{is} \right \}$, where $S_{v_i} \cap S_{v_j}= \emptyset$ for every v _i ≠ v _j. These s new vertices are called copies of v _i. More precisely, $V(G_S)= \left \{v_{11},\ldots ,v_{1s},\ldots ,v_{n1},\ldots ,v_{ns} \right \}$ and $E(G_s)=\left \{ S_{v_i} \cup S_{v_j} : \left \{v_i,v_j \right \} \in E \right \}$. Note that |V (G _s)| = s ⋅|V (G)| and |E(G _s)| = |E(G)|.

For a graph G = (V, E) and an integer k ≥ 2, the k-expansion G ^k of G (also called the k ^th power graph of G) is a k-uniform hypergraph obtained from G by adding k − 2 new vertices of degree one to each edge of G. Note that |V (G ^k)| = |V (G)| + (k − 2) ⋅|E(G)| and |E(G ^k)| = |E(G)|.

Let s ≥ 1 and k ≥ 2s be two integers and consider a graph G. The generalized power graph $G^k_s$ is the k-uniform hypergraph (G _s)^k, obtained by adding k − 2s new vertices to each edge of G _s. These (k − 2s) ⋅|E(G)| new vertices of degree one are called additional vertices of $G^k_s$. Note that $|V(G^k_{s})|=s\cdot |V(G)|+(k-2s)\cdot |E(G)|$ and $|E(G^k_{s})|=|E(G)|$. See an example in Fig. 1.

Fig. 1

Graph G = P ₄ an its respective G ₂ and $G^6_2$

Let G be a simple graph with n vertices. The adjacency matrix of G, denoted by A(G), is the n × n symmetric matrix with entries a _ij = 1 if there is an edge joining vertices v _i and v _j; and a _ij = 0 otherwise. The degree matrix of G, denoted by D(G), is the n × n diagonal matrix defined as D(G) = Diag(d(v ₁), …, d(v _n)) where d(v _i) is the degree of the vertex v _i. The signless Laplacian matrix for G, denoted by Q(G), is the n × n symmetric matrix given by Q(G) = D(G) + A(G). We denote the eigenvalues of A(G) as λ ₁(G) ≥… ≥ λ _n(G) and the eigenvalues of Q(G) as q ₁(G) ≥… ≥ q _n(G).

Let H be a hypergraph with n vertices. The adjacency matrix of H, denoted by A(H) is the n × n symmetric matrix with entries $a_{ij}=| \left \{ e \in E(H):v_i,v_j\in e \right \}|$. We also denote the eigenvalues of A(H) as λ ₁(H) ≥… ≥ λ _n(H).

Note that all previously defined matrices are real and symmetric, so they are Hermitian (a square matrix that is equal to its own conjugate transpose).

Now, we recall some matrix theory results that we use latter. Let X be a m × n matrix and let Y  be a p × q matrix. The kronecker product X ⊗ Y  is the mp × nq matrix:

$$\displaystyle \begin{aligned}X \otimes Y =\left[\begin{array}{ccc} x_{11}Y & \ldots & x_{1n}Y \\ \vdots & \ddots & \vdots \\ x_{m1}Y & \ldots & x_{mn}Y \end{array}\right].\end{aligned}$$

Theorem 1 ([16])

Let X be a n × n matrix and Y  a m × m matrix. If x ₁ ≥… ≥ x _n are the eigenvalues of X and y ₁ ≥… ≥ y _m the eigenvalues of Y , then the nm eigenvalues of X ⊗ Y  are: x ₁ y ₁, …, x ₁ y _m, x ₂ y ₁, …, x ₂ y _m, …, x _n y ₁, …, x _n y _m.

The next theorem, by Weyl [12], is a well known inequality that gives lower and upper bounds for the eigenvalues of a matrix sum.

Theorem 2 ([12])

Let X and Y  be square n × n Hermitian matrices with eigenvalues x ₁ ≥… ≥ x _n and y ₁ ≥… ≥ y _n respectively. If the eigenvalues of the sum Z = X + Y  are z ₁ ≥… ≥ z _n , then x _k + y _n ≤ z _k ≤ x _k + y ₁.

A hypergraph version of the Wilf’s theorem was established in [3] stating a relation between the chromatic number and the largest eigenvalue of its adjacency matrix. This generalization can be restricted to uniform hypergraphs as follows:

Theorem 3 ([3])

Let H be a k-uniform hypergraph, then χ _S(H) ≤ 1 + λ ₁(H).

3 Structural Parameters

Graph parameters can be extended to hypergraphs and most of them in more than one way. In this section we investigate how these parameters behave on the class $G^k_s$ and their relation with the respective parameters of the original graph G.

Let H be a k-uniform hypergraph. A set U ⊆ V (H) is a clique if every subset of U with k elements is an edge of H. The clique number is ω(H) = max{|U| : U ⊆ V (H) is a clique}.

Proposition 1

Given a graph G with at least one edge, s ≥ 1 and k ≥ 2s (except the case where s = 1 and k = 2, i.e. $G^k_{s}=G$ ), we have that $\omega (G^k_{s})=k$ . Moreover, every clique in $G^k_{s}$ is composed by the k vertices of an edge.

Proof

First, observe that the intersection between two edges of $G^k_s$ is formed by a set of s vertices or is empty. Choose any set of k + 1 vertices of $G^k_s$ and suppose it is a clique. This means that there exist two edges in $G^k_s$ which share k − 1 common vertices. This is a contradiction since k ≥ 2s, s ≠ 1 and k ≠ 2. Clearly any set of k vertices of an edge is a clique. □

A matching of a hypergraph H = (V, E) is a set M ⊂ E of pairwise disjoint hyperedges of H. The matching number ν(H) is the cardinality of a maximum matching.

Proposition 2

If G is a graph with s ≥ 1 and k ≥ 2s, then $\nu (G^k_s)=\nu (G)$.

A perfect matching of a hypergraph H is a matching M such that each vertex in V (H) is covered by exactly one edge in M. It is easy to see that for s ≥ 1, G _s has a perfect matching if and only if G has a perfect matching.

Proposition 3

Let G be a graph that is not the union of disjoint edges. For s ≥ 1 and k > 2s the hypergraph $G^k_s$ does not have a perfect matching.

Proof

Since k > 2s, each edge of $G^k_s$ have k − 2s vertices of degree one. The only way that all those vertices are covered by a matching M is when $M=E(G^k_s)$, and that can happen only when G is the union of disjoint edges. □

Given a hypergraph H = (V, E) we construct new hypergraphs by deleting vertices in the following ways. The strong vertex deletion of a vertex v ∈ V  creates the hypergraph H′ = (V ′, E′) where V ′ = V − v and $E'= \left \{ e\in E: v\notin e \right \}$. That is, the strong deletion of v removes v from the vertex set and removes all edges that contain v from the hypergraph. For any subset X of V , we use H −_(S) X to denote the hypergraph formed by strongly deleting all the vertices of X from H. A vertex v ∈ V  is called a strong cut vertex of H if H −_(S) v has more connected components than H, and a set X ⊆ V  is called a strong vertex cut of H if H −_(S) X is disconnected. We define the strong vertex connectivity of H, denoted κ _S(H) as follows: if H has at least one strong vertex cut, then κ _S(H) is the cardinality of a minimum strong vertex cut of H; otherwise, κ _S(H) = |V |− 1. By convention, the strong vertex connectivity of a null or trivial hypergraph is 1. Observe that κ _S(H) ≤ δ(H).

Proposition 4

Given a connected graph G, s ≥ 1 and k ≥ 2s integers such that $G^k_{s} \neq G$ then $\kappa _{S}(G^k_{s})=1$.

Proof

If k > 2s removing a vertex that is originally from G _s disconnects $G^k_s$, since its deletion removes at least one edge and hence the k − 2s additional vertices of this edge become isolated. Similarly, if k = 2s then s > 1 and $G^k_s=G_s$. Removing any vertex leaves the s − 1 vertices that are its copies isolated. □

The weak vertex deletion of a vertex v ∈ V  creates the hypergraph H′ = (V ′, E′) where V ′ = V − v and $E'= \left \{ e-\left \{v\right \}: e\in E \right \}$. That is, the weak deletion of v removes v from the vertex set, and all occurrences of v from the edges of the hypergraph H. For any subset X of V , we use H −_(W) X to denote the hypergraph formed by weakly deleting all the vertices of X from H. Since we are only considering simple hypergraphs, we remove edges with only one vertex. A vertex v ∈ V  is called a weak cut vertex of H if H −_(W) v has more connected components than H, and a set X ⊆ V  is called a weak vertex cut of H if H −_(W) X is disconnected. We define the weak vertex connectivity of H, denote κ _W(H) as follows: if H has at least one weak vertex cut, then κ _W(H) is the cardinality of a minimum weak vertex cut of H; otherwise, κ _W(H) = |V |− 1. By convention, the weak vertex connectivity of a null or trivial hypergraph is 1.

Proposition 5

Given a connected graph G that is not the complete graph and an integer s ≥ 1, then κ _W(G _s) = s.κ(G).

Proof

Note that by the construction of G _s, we have that if X ⊂ V (G _s) is a weak vertex cut of G _s then $X= S_{v_1}\cup S_{v_2} \ldots \cup S_{v_r}$ and {v ₁, …, v _r}⊆ V (G) is a vertex cut of G. Now, let {v ₁, …, v _r} be a minimum vertex cut of G. So, $S_{v_1}\cup S_{v_2} \ldots \cup S_{v_r}$ is a minimum weak vertex cut in G _s with s.κ(G) elements, otherwise v ₁, …v _r would not be a minimum vertex cut of G, a contradiction. □

Proposition 6

Let G be a connected graph that is not the complete graph, s ≥ 1 and k > 2s be two integers. Then:

• (i) If κ _W(G _s) = s, then $\kappa _{W}(G^k_s)=s$.

• (ii) If κ _W(G _s) ≥ 2s, then $\kappa _{W}(G^k_s)=2s$.

Proof

First we observe that a vertex cut of G _s is a vertex cut of $G^k_s$. Also, after the k-expansion $G^k_s$ of G _s, the only new minimum vertex cut is the one where we isolate the additional k − 2s new vertices of an edge by removing the 2s already existing vertices (since the new vertices of $G^k_s$ make no difference in a vertex cut). Hence:

• (i) if X is a minimum weak vertex cut of G _s with less than 2s elements, then it is a minimum weak vertex cut of $G^k_s$.

• (ii) if κ _W(G _s) ≥ 2s, a minimum weak vertex cut of G _s has more than 2s elements. For each edge of $G^k_s$, the set of the 2s vertices that came from G ^s is a minimum weak vertex cut of $G^k_s$ since their removal leaves the additional k − 2s remaining vertices isolated. □

Next result follows from the fact that if s = 1 then G _s = G and $G^k_s=G^k$.

Corollary 1

Let G be a connected graph. For any k > 2 we have that:

• (i) if κ(G) = 1, then κ _W(G ^k) = 1;

• (ii) if κ(G) ≥ 2, then κ _W(G ^k) = 2.

We observe from the previous results that the difference between weak and strong vertex connectivity of hypergraphs can be arbitrarily large, since $\kappa _{S}(G^k_{s})=1$ and $\kappa _{W}(G^k_s)\geq s$, with s as large as desired. Finally, we also remark that the inequality κ _W(H) ≤ δ(H) is not valid: if G is a connected graph with κ(G) ≥ 2, we have for k > 2s that $\delta (G^k_s)=1<2s=\kappa _{W}(G^k_s)$.

The distance d(v, u) between two vertices v and u is the minimum length of a path that connects v and u. The diameter d(H) of H is defined by $d(H)=max \left \{ d(v,u): v,u\in V \right \}$. It is easy to see that given a graph G and s ≥ 1, then d(G _s) = d(G). But this is not always true for the k-expansion.

Proposition 7

$d(G) \leq d(G^k_{s}) \leq d(G)+2$ , for any graph G, s ≥ 1 and k ≥ 2s, .

Proof

Suppose d(G _s) = r and P = v ₁, e ₁, v ₂, e ₂, …, v _r, e _r, v _r+1 be a maximum path of G _s. If k > 2s, we add k − 2s vertices on each edge to obtain $G^k_s$. After that, if there is an additional vertex u such that $\left \{u,v_1 \right \}$ belongs to an edge e ≠ e ₁ and another additional vertex w such that $\left \{ w,v_{r+1} \right \}$ belongs to an edge f ≠ e _r, the path P = u, e, v ₁, e ₁, v ₂, e ₂, …, v _r, e _r, v _r+1, f, w have length d(G) + 2. Moreover, $d(G^k_{s})=d(G)+2$ since we have at most 2 additional vertices on a path and the path must start and end on them, otherwise we would have to repeat edges. □

A hypergraph coloring is an assigning of colors $\left \{1, 2,\ldots , c\right \}$ to each vertex of V (H) in such a way that each edge contains at least two vertices of distinct colors. A coloring using at most c colors is called a c-coloring. The chromatic number χ(H) of a hypergraph H is the least integer c such that H has a c-coloring.

It is easy to see that given a graph G we have that $\chi (G^k)=\chi (G_s)=\chi (G^k_s)=2$ (except when s = 1 and k = 2). Another type of coloring, that is also a generalization of graph coloring, is the strong hypergraph coloring: is an assigning of colors $\left \{1, 2,\ldots , c\right \}$ to each vertex of V (H) in such a way that every vertex of an edge has distinct colors. The strong chromatic number χ _S(H) of a hypergraph H is the least integer c such that H has a strongly c-coloring. Given a hypergraph H, note that:

• 1. χ _S(H) ≥|e| for every e ∈ E(H);

• 2. χ(H) ≤ χ _S(H), since a strong hypergraph coloring is also a hypergraph coloring;

• 3. ω(H) ≤ χ _S(H) (similarly to graphs);

• 4. for the class $G^k_s$, the inequality ω(H) ≤ χ(H) is not valid, since $\chi (G^k_s)=2$ but we can have edges (cliques) arbitrarily large.

We do not consider $\chi (G)=\chi _S(G^k_{s})=1$, since G has at least one edge. The following results establish relations between χ(G), χ _S(G _s) and $\chi _S(G^k_{s})$.

Proposition 8

If G is a graph and s ≥ 1 is an integer, then χ _S(G _s) ≤ s.χ(G).

Proof

Let χ(G) = c, we obtain a sc-strong coloring of G _s as follows: if v ∈ V (G) has color c(v) ∈{1, …, c} then, in G _s, assign colors {1 + (c(v) − 1)s, 2 + (c(v) − 1)s, …, s + (c(v) − 1)s} to S _v. □

Note that this bound is tight in the sense that the equality holds for any s-extension of the complete graph and does not hold for the 2-extension of C ₅.

Proposition 9

Let s ≥ 1, k > 2s be two integers and let G be a graph. We have that:

• (i) if χ _S(G _s) < k then $\chi _S(G^k_{s})=k$;

• (ii) if χ _S(G _s) ≥ k then $\chi _S(G^k_{s})=\chi _S(G_s)$.

Proof

• (i) Let χ _S(G _s) = c < k, we obtain a k-strong coloring of $G^k_{s}$ as follows: we color the vertices of $G^k_{s}$ that came from G _s with the same c-colors used in G _s. Hence, for each edge of $G^k_s$, we already used 2s colors from the set $\left \{1,2,..,k \right \},\ k>2s$. Again, for each edge, we color the k − 2s new additional vertices with the remaining k − 2s distinct colors. This k-strong coloring of $G^k_s$ is minimum, since k is the size of each edge of $G^k_s$.

• (ii) Let χ _S(G _s) = c ≥ k and consider a c-strong coloring of G _s. We color the vertices of $G^k_{s}$ that came from G _s with the same c-colors used in G _s. For each edge, we color the k − 2s additional vertices with any k − 2s distinct colors from $\left \{ 1,2,..,c \right \}$ different from the 2s colors already used in the vertices that came from G _s (since c ≥ k ≥ 2s such colors exist). Suppose that it is possible to use less than c-colors in $G^k_{s}$. This implies that we can color all the vertices of $G^k_{s}$ that came from G _s with less than c-colors and hence G _s with less than c colors, a contradiction. □

Corollary 2

Let G be a graph and k ≥ 2 an integer. Thus:

• (i) if χ(G) < k then χ _S(G ^k) = k;

• (ii) if χ(G) ≥ k then χ _S(G ^k) = χ(G).

A set U ⊆ V  is a strong independent set if no two vertices of U are adjacent. The strong independence number is α′(H) = max{|U| : U ⊆ V (H) is a strong independent set of H }. Let G be a graph and s ≥ 1. From the construction of G _s we have that α′(G _s) = α(G).

Proposition 10

If G is a graph, s ≥ 1 and k > 2s, then $\alpha '(G^k_s)=|E(G)|$.

Proof

Since k > 2s, every edge of $G^k_s$ has at least one additional vertex. A set formed by choosing, for each edge, one of these additional vertices is a strong independent set of size $|E(G^k_s)|=|E(G)|$. This set is maximum since α′(H) ≤|E(H)|, for any hypergraph H. □

Another generalization of a graph independent set is as follows: a set U ⊆ V  is an independent set if no edge of H is contained in U. As before, the independence number is α(H) = max{|U| : U ⊆ V (H) is an independent set of H }. Observe that if U is a strong independent set of a hypergraph H then U is also an independent set of H, since if U contains no two adjacent vertices then U does not contain an edge of H. So we have that α′(H) ≤ α(H).

Proposition 11

If G is a graph and s ≥ 1, then α(G _s) ≥ (s − 1) ⋅|V (G)| + α(G).

Proof

Let V (G) = {v ₁, …, v _n} and $V(G_s)=S_{v_1} \cup \ldots \cup S_{v_n}$. We obtain an independent set with (s − 1) ⋅ n elements by choosing s − 1 vertices of $S_{v_i}$, for each i = 1, …, n. Now, adding a maximum stable set of G to the previous set produces a stable set of G _s with (s − 1) ⋅ n + α(G) vertices. □

Proposition 12

If G be a graph, s ≥ 1 and k ≥ 2s, then $\alpha (G^k_s)\geq (s-1)\cdot |V(G)|+\alpha (G)+(k-2s)\cdot |E(G)|$.

Proof

By the construction of $G^k_s$ and Proposition 11, a stable set of G _s is also a stable set of $G^k_s$ with (s − 1) ⋅|V (G)| + α(G) vertices. Adding to this stable set every k − 2s additional vertices of each edge of $G^k_s$ produces a stable set with (s − 1) ⋅|V (G)| + α(G) + (k − 2) ⋅|E(G)| elements. □

Corollary 3

Let G be a graph and k ≥ 2, then α(G ^k) ≥ α(G) + (k − 2) ⋅|E(G)|.

4 Spectral Parameters

In this section we investigate spectral properties of hypergraphs and establish relations with structural parameters. The following result relates the adjacency matrix of G _s with the matrices A(G) and Q(G).

Proposition 13

Let G be a graph with n vertices and s > 1. The adjacency matrix A(G _s) is given on s × s blocks of size n × n by:

$$\displaystyle \begin{aligned}A(G_s) = \left[\begin{array}{ccccc} A(G) & Q(G) & Q(G) & \ldots & Q(G) \\ Q(G) & A(G) & Q(G) & \ldots & Q(G) \\ Q(G) & Q(G) & A(G) & \ldots & Q(G) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ Q(G) & Q(G) & Q(G) & \ldots & A(G) \end{array}\right]=(J_{s} \otimes Q(G))+(I_s \otimes -D(G)) ,\end{aligned}$$

where J _s is the s × s matrix with 1 on all entries and I _s is the s × s identity matrix.

Proof

First we show that A(G _s) can be written in blocks like above. Let G be a graph on n vertices, then |V (G _s)| = sn. So, we order the vertices of the matrix as follows: V (G _s) = {v ₁₁, v ₂₁, …, v _n1, v ₁₂, v ₂₂, …, v _n2, v ₁₃, v ₂₃, …, v _n3, …, v _1s, v _2s, …, v _sn}, where $S_{v_1}=\left \{v_{11},v_{12},\ldots ,v_{1s} \right \},\ S_{v_2}=\left \{v_{21},v_{22},\ldots ,v_{2s} \right \}$, …, $S_{v_n}=\left \{v_{n1},v_{n2},\ldots ,v_{ns} \right \}$. We suppose that the vertices v ₁₁, v ₂₁, …, v _n1 are the vertices that come from G. So the n × n block formed by these is A(G), since two vertices that are not copies from each other, share an edge in G _s if and only if they share an edge in G. Hence, we can see that all the diagonal blocks, formed by the vertices $\left \{v_{1i},v_{2i},\ldots ,v_{ni} \right \} \times \left \{v_{1i},v_{2i},\ldots ,v_{ni} \right \}$ , i = 1, …, s, also correspond to A(G).

For the other blocks we observe that, for every i ≠ j, the blocks formed by $\left \{v_{1i},v_{2i},\ldots ,v_{ni} \right \} \times \left \{v_{1j},v_{2j},\ldots ,v_{nj} \right \}$ are always the same, since the vertices are copies from one another.

The block where i = 1 and j = 2 have the following structure: the vertices v ₁₁ and v ₁₂ are copies so they are in the same edges; and the number of edges they belong is exactly d _G(v ₁). So, their entry is equal d _G(v ₁), the degree of v ₁ in G. The same works for the entries v ₂₁ × v ₂₂, v ₃₁ × v ₃₂,…,v _n1 × v _n2. So, the diagonal of the block is made of the degrees in G. The entries that are not in the diagonal, for example, the entry v ₁₁ × v22 is the same entry as v ₁₁ × v ₂₁, since v ₂₂ is a copy of the vertex v ₂₁. So, these blocks are equal D(G) + A(G) = Q(G). □

Proposition 14

Let G be a graph on n vertices, s > 1 an integer and d ₁, …, d _n the vertices degree of G. Then − d ₁, …, −d _n are eigenvalues of A(G _s). Moreover, each − d _i has multiplicity at least s − 1.

Proof

Consider the vector (−1, 0, …, 0|, 1, 0, …, 0|, 0, …, 0|, …, |0, …, 0) ∈ R ^sn, formed of s “blocks” with n entries each (ie, |− 1, 0, …, 0| has n entries, |1, 0, …, 0| has n entries, |0, …, 0| has n entries). This vector is an eigenvector of A(G _s) associated to the eigenvalue − d ₁. Indeed:

$$\displaystyle \begin{aligned}\left[\begin{array}{ccccc} A(G) & Q(G) & Q(G) & \ldots & Q(G) \\ Q(G) & A(G) & Q(G) & \ldots & Q(G) \\ Q(G) & Q(G) & A(G) & \ldots & Q(G) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ Q(G) & Q(G) & Q(G) & \ldots & A(G) \end{array}\right]. \left[\begin{array}{c} -1 \\ 0 \\ \vdots \\ 0 \\ --\\ 1 \\ 0 \\ \vdots \\ 0 \\ --\\ 0 \\ \vdots \\ 0 \end{array}\right]= \left[\begin{array}{c} -1.0+1.d_1 \\ -a_{2,1}+a_{2,1} \\ \vdots \\ -a_{n,1}+a_{n,1} \\ --\\ -1.d_1+1.0 \\ -a_{2,1}+a_{2,1} \\ \vdots \\ -a_{n,1}+a_{n,1}\\ --\\ 0 \\ \vdots \\ 0 \end{array}\right]= \left[\begin{array}{c} d_1 \\ 0 \\ \vdots \\ 0 \\ --\\ -d_1 \\ 0 \\ \vdots \\ 0 \\ --\\ 0 \\ \vdots \\ 0 \end{array}\right]\end{aligned}$$

Note that the vectors

(−1, 0, …, 0|, 0, …, 0|, 1, 0, …, 0|, …, |0, …, 0),

(−1, 0, …, 0|, 0, …, 0|, 0, …, 0|, 1, 0, …, 0|, …, |0, …, 0), …,

(−1, 0, …, 0|, 0, …, 0|, 0, …, 0|, …, |1, …, 0)

are also eigenvectors of A(G _s) associated to the eigenvalue − d ₁. Since we have s blocks, the multiplicity of − d ₁ is at least s − 1. Similarly to − d ₂, starting with the eigenvector:

(0, −1, …, 0|, 0, 1, …, 0|, 0, …, 0|, …, |0, …, 0)

up to − d _n, when starting with the eigenvector:

(0, 0, …, −1|, 0, 0, …, 1|, 0, …, 0|, …, |0, …, 0). □

Next result immediately follows from the previous proposition observing that if G is connected then every vertex degree is positive.

Corollary 4

If G is a graph on n vertices and s > 1 an integer, then A(G _s) has at least n ⋅ (s − 1) non positive eigenvalues. Moreover, if G is connected then A(G _s) has at least n ⋅ (s − 1) negative eigenvalues (hence, A(G _s) has at most n non negative eigenvalues).

Next proposition provides bounds for the greatest eigenvalue of A(G _s).

Proposition 15

If G be a graph with n vertices and s > 1 integer, then

$$\displaystyle \begin{aligned}s.q_1(G)-\varDelta(G) \leq \lambda_1(G_s) \leq s.q_1(G)-\delta(G).\end{aligned}$$

Proof

For the left inequality, we observe that is known that the largest eigenvalue of J _s is s. Thus, by Theorem 1, the largest eigenvalue of J _s ⊗ Q(G) is s ⋅ q ₁(G). Also, the smallest eigenvalue of − D(G) is − Δ(G). So, by Theorem 1, the smallest eigenvalue of I _s ⊗−D(G) is − Δ(G). Since A(G _s) = (J _s ⊗ Q(G)) + (I _s ⊗−D(G)), from Theorem 2, we have that s ⋅ q ₁(G) − Δ(G) ≤ λ ₁(G _s).

For the right inequality, we observe again that the largest eigenvalue of J _s ⊗ Q(G) is s.q ₁(G). Also, the largest eigenvalue of − D(G) is − δ(G). So, by Theorem 1, the largest eigenvalue of I _s ⊗−D(G) is − δ(G). Since A(G _s) = (J _s ⊗ Q(G)) + (I _s ⊗−D(G)), we have from theorem 2 that λ ₁(G _s) ≤ s.q ₁(G) − δ(G). □

We observe that the bound given by Proposition 15 is tight in the sense that the equality holds for any regular graph G and for any s > 1. In what follows we obtain some results relating structural and spectral parameters.

A well known spectral graph theory result is: given a connected graph G the number of distinct eigenvalues of A(G) is at least d(G) + 1 (this is also true for the number of distinct eigenvalues of Q(G)). This result is still true on hypergraphs, and the proof is basically the same. In [5] this bound is proved for the signless Laplacian matrix of a hypergraphs. We prove this result for hypergraphs adjacency matrix but first we present the following lemma.

Lemma 1

Let H be a hypergraph and A = A(H) its adjacency matrix. (A ^l)_i,j > 0 if there is a path with length l connecting two distinct vertices i and j, and (A ^l)_i,j = 0 otherwise (where (A ^l)_i,j denotes the entry i, j of A(H)^l ).

Proof

The proof is by induction on l. If l = 1 the property clearly holds. Suppose the statement is true for l ≥ 1 and now we check for l + 1. Note that $(A^{l+1})_{i,j}= \sum _{k=1}^n (A^l)_{i,k}(A)_{k,j}$. If there is a path with length l + 1 joining i and j then there must exist a path with length l joining i to a neighbor u of j. So (A)_u,j = 1 and by induction hypothesis (A ^l)_i,u > 0. Therefore (A ^l+1)_i,j > 0. If there is no path with length l + 1 joining i and j then there does exist no path with length l joining i to any neighbor of j. So, if u is a neighbor of j we have that (A ^l)_i,u = 0. When u is not a neighbor of j, we have that (A)_u,j = 0. Therefore (A ^l+1)_i,j = 0. □

Proposition 16

If H is a connected hypergraph then $|\left \{\mathit{\mbox{distinct eigenvalues of}}\right .$ $\left .\!A(H)\right \}|\ \geq d(H)+1.$

Proof

Let λ ₁, …, λ _t be all the distinct eigenvalues of A = A(H). Then (A − λ ₁ I)…(A − λ _t I) = 0. So, we have that A ^t is a linear combination of A ^t−1, …, A, I. Suppose by contradiction that t ≤ d(H). Hence there exist vertices i and j such that d(i, j) = t and from our previous lemma, we have that (A ^t)_i,j > 0. since there exists no path with length shorter than t joining i and j, (A ^t−1)_i,j = 0, …, (A)_i,j = 0, (I)_i,j = 0. This is a contradiction, since (A ^t)_i,j = c ₁(A ^t−1)_i,j + … + c _t−1(A)_i,j + c _t(I)_i,j. □

Previous proposition together with Proposition 7 result this simple corollary.

Corollary 5

If G is connected then $|\left \{\mathit{\mbox{distinct eigenvalues of}}~A(G^k_s) \right \}| \geq d(G)+1$.

In other words, to find connected hypergraphs of the class $G^k_s$ with few distinct adjacency eigenvalues, we have to look for graphs G with small diameter.

The next proposition gives us a different bound for $\chi _S(G^k_s)$, in terms of the largest eigenvalue of Q(G) and the minimum degree of the graph G.

Proposition 17

Given a graph G, s > 1 and k ≥ 2s we have that $\chi _S(G^k_s)=k$ or $\chi (G^k_s) \leq 1+s.q_1(G)-\delta (G)$.

Proof

If $\chi _S(G^k_{s}) \neq k$ then by Proposition 9 $\chi _S(G^k_{s})=\chi _S(G_{s})$. Where by Theorem 3 and Proposition 15 we have: χ _S(G _s) ≤ 1 + λ ₁(G _s) ≤ 1 + s.q ₁(G) − δ(G). □

A result from spectral graph theory states that if G is a graph, then $\alpha (G) \leq min \left \{\lambda (G)^-,\lambda (G)^+\right \}$, where λ(G)⁻ is the number of non positive eigenvalues of A(G) and λ(G)⁺ is the number of non negative eigenvalues of A(G)”. We show that this is not valid for the independence number of the class G _s.

Proposition 18

If s > 1 and G is a connected graph on n vertices, then $\alpha (G_s) > min\left \{\lambda (G_s)^-,\lambda (G_s)^+\right \}$.

Proof

From Corollary 4, we have that A(G _s) has at most n non negative eigenvalues. Hence, by Proposition 11: $\alpha (G_s) \geq (s-1)n+\alpha (G) > n \geq \lambda (G_s)^+ \geq min \left \{\lambda (G_s)^-,\lambda (G_s)^+\right \}$. □

Another result states that, for any graph $G,\ \frac {|V(G)|}{\alpha (G)} \leq \lambda _1(G)+1$”. This fact has not yet been generalized for hypergraphs and we prove its validity for connected hypergraphs in the class G _s.

Proposition 19

If G is connected on n vertices and s > 1 then $\frac {|V(G_s)|}{\alpha (G_s)} \leq \lambda _1(G_s)+1$.

Proof

By Proposition 11, we have that $\frac {|V(G_s)|}{\alpha (G_s)} = \frac {sn}{\alpha (G_s)} \leq \frac {sn}{(s-1)n+\alpha (G)} \leq \frac {sn}{(s-1)n}=\frac {s}{s-1}$. From Proposition 15, we have that s ⋅ q ₁(G) − Δ(G) ≤ λ ₁(G _s). Thus, it suffices to show that $\frac {s}{(s-1)} \leq s.q_1(G)-\varDelta (G)+1$ or, in other words, that s ≤ (s − 1)(s.q ₁(G) − Δ(G) + 1). Since s > 1, if s.q ₁(G) − Δ(G) + 1 ≥ 2 then the above inequality is valid, indeed: s.q ₁(G) −Δ(G) + 1 ≥ s(Δ(G) + 1) −Δ(G) + 1 = (s− 1)Δ(G) + s + 1 ≥ 2. Where the first inequality holds because: [6] If G is a connected graph then q ₁(G) ≥ Δ(G) + 1. □