Some New Families of Noncorona Graphs with Strong Anti-Reciprocal Eigenvalue Property

Abstract

Let G be a simple graph and A(G) be its adjacency matrix. We say that the graph G satisfies reciprocal eigenvalue property (or simply, property (R)) if $\dfrac{1}{\lambda }$ is an eigenvalue of A(G), whenever $\lambda$ is an eigenvalue of A(G). The graph G satisfies strong anti-reciprocal property (or simply, property (-SR)) if $-\dfrac{1}{\lambda }$ is an eigenvalue of A(G) with multiplicity k, whenever $\lambda$ is an eigenvalue of A(G) with multiplicity k. Recently, Barik, Mondal and Pati proved that there exists no singular (non-trivial) tree whose nonzero eigenvalues satisfy reciprocal eigenvalue property. As a concluding remark, the authors mentioned that “It is interesting to see that our result (for trees) is true for any connected graph”. In this paper, we prove that there exists no singular unicyclic graph whose nonzero eigenvalues satisfy reciprocal eigenvalue property. Ahmad, Hameed and Jabeen in their recent paper raised the question about existence of noncorona graphs with property (-SR) and constructed seven classes of graphs satisfying this property. Here, we give several families of noncorona graphs with property (-SR).

1 Introduction

All the graphs considered in this paper are simple. Let G be a graph on n vertices and A(G) be its adjacency matrix. We denote the eigenvalues of A(G) by $\lambda _{1}, \lambda _{2},\ldots ,\lambda _{n}$ where $\lambda _{1}\ge \lambda _{2}\ge \cdots \ge \lambda _{n}$. The eigenvalues of A(G) are called the eigenvalues of graph G. The characteristic polynomial of G is the characteristic polynomial of A(G). It is denoted by $p(G;x)=\sum _{i=0}^{n}a_{i}x^{n-i}$. The graph G is non-singular if all its eigenvalues are nonzero. Otherwise, G is called a singular graph. The number of zero eigenvalues of the graph G is called its nullity. A non-singular graph G satisfies property (R) if and only if $\dfrac{1}{\lambda _{i}}$ is an eigenvalue of G for $1\le i\le n$. Furthermore, if both $\lambda _{i}$ and $\dfrac{1}{\lambda _{i}}$ are the eigenvalues of G with same multiplicity, then this property is referred as strong reciprocal property (SR) of G. The study on graphs with property (R) was initiated by Barik et al. [10]. The property (SR) of trees was investigated under the names symmetric property and property C, respectively, by Godsil and Mckay [14], and by Cvetković et al. [13] in the year 1978. In literature, the existence of bipartite graphs for which the property (R) and the property (SR) are not equivalent is still an open problem. Classes of nonbipartite graphs with property (R) but not (SR) are presented in [9]. As a variation of property (R) and property (SR), Lagrange [17] introduced property (-R) and (-SR) for graphs in the year 2012. The anti-reciprocal property (simply, property (-R)) is satisfied by G if and only if $-\dfrac{1}{\lambda _{i}}$ is an eigenvalue of G for $1\le i\le n$. Also, graph G satisfies strong anti-reciprocal property (property (-SR)) if and only if both $\lambda _{i}$ and $-\dfrac{1}{\lambda _{i}}$ are the eigenvalues of G with same multiplicity. Note that the eigenvalues of a bipartite graph are symmetric about the origin. Thus, the property (R) (resp. property (SR)) and the property (-R) (resp. property (-SR)) are same for a bipartite graph. Corona product [15, 18] of two graphs G and H, denoted by $G\circ H$, is the graph obtained by taking one copy of G and n copies of H, and then joining each vertex of G with the corresponding copy of H. If $H=K_{1}$, then the graph $G\circ H$ is called simple corona graph. In [13, 14], it is proved that a non-singular tree satisfies property (SR) if and only if it is a simple corona graph, and in [8], the authors proved that for non-singular trees, the property (R) and property (SR) are equivalent. Characterization of unicyclic graphs by property (SR) has been done in [7] by Barik, Nath, Pati and Sarma. In [5], nonbipartite unicyclic graphs are studied for property (-SR), and it is shown that they are necessarily simple corona graphs. Studies on weighted graphs with property (SR) and property (-SR) can be seen in [2, 4].

In [6], the authors proved that there exist no (non-trivial) trees which has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property, and as a remark, the authors pose a more general question that “Whether their result is true for any connected graph which has at least one zero eigenvalue?". Motivated by this, in Sect. 2 of our paper, we prove that there is no singular unicyclic graph whose nonzero eigenvalues satisfy the reciprocal eigenvalue property. In [1], the authors raised the question about existence of noncorona graphs with property (-SR) and constructed seven classes of graphs satisfying this property. Later, in [5], all these constructions were generalized by Barik, Ghosh and Mondal. With this motivation, in Sect. 3, we give several families of noncorona graphs with property (-SR)

1.1 Notations

The following notations are used in the subsequent sections. We denote a column vector of size n with all its entries equal to one by $\mathbf {1_{n}}$ or simply, ${\textbf {1}}$. A $n\times m$ matrix of all ones is denoted by $J_{n\times m}$ or J. The i-th entry of the column vector $e_{i}$ is equal to one and its all other entries are zero. The matrix $E_{ii}$ is defined as $e_{i}e^{T}_{i}$. Let $e^{i}_{4}$ be a column vector of size four having its i-th entry as one and all other entries as zero. The null matrix of order $n\times m$ is denoted by ${\textbf {0}}_{n\times m}$. Let $C_{2p_{i}}$ ($p_{i}\ge 2$) be a cycle graph of order $2p_{i}$. Let $\{v_{i1},v_{i2},\ldots ,v_{ip_{i}}\}$ and $\{u_{i1},u_{i2},\ldots ,u_{ip_{i}}\}$ be the two disjoint vertex independent sets of the cycle $C_{2p_{i}}$. Define $H_{i}=C_{2p_{i}}\circ K_{1}$ and denote the pendent vertices of $H_{i}$ that are adjacent with the vertices $v_{it}$ and $u_{it}$ ($t=1,2,\ldots ,p_{i}$) by $v^{\prime }_{it}$ and $u^{\prime }_{it}$, respectively. Let $A(H_{i})=\left[ \begin{array}{cc} {{\textbf {0}}}_{2p_{i}\times 2p_{i}}&{}I_{2p_{i}}\\ I_{2p_{i}}&{}\begin{array}{cc} {{\textbf {0}}}_{p_{i}\times p_{i}}&{}C\\ C^{T}&{}{{\textbf {0}}}_{p_{i}\times p_{i}} \end{array} \end{array}\right]$ where $\left[ \begin{array}{cc} {{\textbf {0}}}_{p_{i}\times p_{i}}&{}C\\ C^{T}&{}{{\textbf {0}}}_{p_{i}\times p_{i}} \end{array}\right]$ is the adjacency matrix of $C_{2p_{i}}$. For a cycle graph $C_{m_{j}}$ of order $m_{j}$, define $F_{j}=C_{m_{j}}\circ K_{1}$.

2 Unicyclic Graphs with Reciprocal Eigenvalue Property

A linear subgraph of G is a collection of vertex disjoint union of cycles and edges of G. A matching of G is a collection of independent edges of G. A matching is a perfect matching if the matching edges covers all the vertices of G. In this section, we prove that there is no singular unicyclic graph whose nonzero eigenvalues satisfies property (SR).

The Kronecker product of two matrices $A=(a_{ij})_{n\times m}$ and $B=(b_{ij})_{p\times q}$, denoted by $A\otimes B$, is the matrix of order $np\times mq$, obtained from A and B, by replacing each entry $a_{ij}$ of A by $a_{ij}B$. It can be verified that $(A\otimes B)(C\otimes D)=AC\otimes BD$. As a consequence, if $\lambda$ and $\mu$ are the eigenvalues of A and B corresponding to the eigenvectors X and Y, respectively, then $\lambda \mu$ is an eigenvalue of $A\otimes B$ corresponding to the eigenvector $X\otimes Y$. For more details, see [16].

In [6], the authors proved that if G satisfies property (R), then G is unimodular, i.e., $|\det (A(G))|=1$. In a similar way, we prove the following result.

Lemma 2.1

Let G be a graph with nullity k such that its nonzero eigenvalues satisfy (anti-) reciprocal eigenvalue property. Then $|a_{n-k}|=1$.

Proof

Suppose that the nonzero eigenvalues of G satisfies anti-reciprocal eigenvalue property. Let $m=a_{n-k}$. Then $m={(-1)^{n-k}}\lambda _{1}(G)\lambda _{2}(G)\cdots \lambda _{n-k}(G)$. Since $-\dfrac{1}{\lambda _{i}(G)}$ ($1\le i\le n-k$) is an eigenvalue of G, $\dfrac{1}{m}=\dfrac{(-1)^{n-k}}{\lambda _{1}(G)\lambda _{2}(G)\cdots \lambda _{n-k}(G)}$ is an eigenvalue of the matrix $A_{n-k}:=\underbrace{A(G)\otimes A(G)\otimes \cdots \otimes A(G)}_{n-k~ \text {times}}$. Thus, the rational number $\dfrac{1}{m}$ is a root of a monic polynomial over ${\mathbb {Z}}$, namely the polynomial $\det (A_{n-k}-xI)$. Proving that $m=\pm 1$, i.e., $|a_{n-k}|=1$. Similarly, one can easily prove that the result is true if the nonzero eigenvalues of G satisfy reciprocal eigenvalue property. $\square$

The following lemma is used to prove our next result.

Lemma 2.2

Let G be a connected graph with nullity $k>0$. Then the number of matchings of order $n-k$, if exists, is at least 2.

Proof

Suppose that H is the unique matching of G of order $n-k$. Since $k>0$, there is a vertex u of G which is not in H, and since G is connected u must be adjacent with some vertex v in G. Now, if v is an end vertex of some edge e in H, then $H\backslash \{e\}\cup \{uv\}$ is a matching of order $n-k$, a contradiction. Otherwise, the vertex v is not in H. In this case, $H\backslash \{e\}\cup \{uv\}$, where e is any edge component of H, is a matching of order $n-k$, again a contradiction. $\square$

Theorem 2.3

Let G be a unicyclic graph of order n with nullity $k>0$. Then $|a_{n-k}|>1$.

Proof

Let C be the unique cycle of G of length $\ell$. Let $p(G, x)=\sum _{i=0}^{n}a_{i}x^{n-i}$ be the characteristic polynomial of G. Since nullity of G is k ($>0$), we have $a_{n-k}\ne 0$ and $a_{i}=0$ for $i>n-k$. Assume to the contrary that $|a_{n-k}|=1$.

From Sachs theorem [12], we have

$$\begin{aligned} a_{i}=\sum _{H\in {\mathcal {H}}_{i}}(-1)^{w(H)}2^{c(H)}, \end{aligned}$$

where ${\mathcal {H}}_{i}$ is the collection of all linear subgraphs (vertex disjoint union of edges and cycles) of G of order i, w(H) is the number of edges and cycles in the linear subgraph H, and c(H) is the number of cycle components in H.

First we assume that G is an odd unicyclic graph. If e(H) is the number of edge components in a linear subgraph H of G and ${\mathcal {M}}(G)_{i}$ is a matching of G of order i. Then

$$\begin{aligned} a_{n-k}=\sum _{H\in {\mathcal {M}}(G)_{_{n-k}}}(-1)^{e(H)}\quad \text {or}\quad a_{n-k}=-2\sum _{H\in {\mathcal {M}}(G\backslash C)_{n-k-\ell }}(-1)^{e(H)}, \end{aligned}$$

depending upon whether $n-k$ is even or odd, respectively. Therefore, if $n-k$ is odd, then $|a_{n-k}|\ge 2$, a contradiction to our assumption. Otherwise, $n-k$ is even. Then

$$\begin{aligned} a_{n-k}=\sum _{H\in {\mathcal {M}}(G)_{n-k}}(-1)^{e(H)}. \end{aligned}$$

By our assumption, $|a_{n-k}|=1$. Therefore, $|{\mathcal {M}}(G)_{n-k}|=1$, a contradiction to Lemma 2.2.

Next we assume that G is a unicyclic graph with even cycle. Therefore,

$$\begin{aligned} a_{n-k}=\sum _{H\in {\mathcal {M}}(G)_{n-k}}(-1)^{e(H)}-2\sum _{H\in {\mathcal {M}}(G\backslash C)_{n-k-\ell }}(-1)^{e(H)}. \end{aligned}$$

Now, if $\ell /2$ is odd, then

$$\begin{aligned} |a_{n-k}|=\sum _{H\in {\mathcal {M}}(G)_{n-k}}1+\sum _{H\in {\mathcal {M}}(G\backslash C)_{n-k-\ell }}2. \end{aligned}$$

(2.1)

Since $|a_{n-k}|=1$, by (2.1), it follows that G has a unique matching of order $n-k$, a contradiction to Lemma 2.2. Otherwise, $\ell /2$ is even. In this case, we have

$$\begin{aligned} |a_{n-k}|=\sum _{H\in {\mathcal {M}}(G)_{n-k}}1-\sum _{H\in {\mathcal {M}}(G\backslash C)_{n-k-\ell }}2. \end{aligned}$$

(2.2)

Since the cycle C give rise to two matchings of order $\ell$, Eq. (2.2) reduces to

$$\begin{aligned} |a_{n-k}|=\sum _{H\in \mathcal {M^*}(G)_{n-k}}1, \end{aligned}$$

where $M^{*}(G)_{n-k}$ is a matching of G of order $n-k$, not containing any of the two perfect matchings of the cycle C.

Since $|a_{n-k}|=1$, $|M^{*}(G)_{n-k}|=1$, i.e., there is an unique matching $H^{*}$, say, of G with $n-k$ vertices, not containing any of the two perfect matchings of the cycle C. As $k>0$, there is a vertex u of G which is not in $H^{*}$, and since G is connected, u is adjacent with some vertex v, say, of G. If the vertex v is not in $H^{*}$, then we can choose an edge e of $H^{*}$ such that $H^{*}\backslash \{e\}\cup \{uv\}\in M^{*}(G)_{n-k}$, a contradiction. Otherwise, v is an end vertex of some edge e of $H^{*}$. In this case, $H_{1}:=H^{*}\backslash \{e\}\cup \{uv\}$ is a matching of order $n-k$. If $H_{1}\notin M^{*}(G)_{n-k}$, then $H_{1}$ contains a perfect matching of the cycle C. Let $M_{1}=\{e_{1}=uv, e_{2}, \ldots , e_{\ell /2}\}$ be the perfect matching of the cycle C contained in $H_{1}$ and $M_{2}=\{e^{\prime }_{1}=vx, e^{\prime }_{2}, \ldots , e^{\prime }_{\ell /2}\}$ be another perfect matching of the cycle C. Then $H_{1}^{\prime }:=(H_{1}\backslash M_{1})\cup (M_{2}\backslash e^{\prime }_{1})\cup \{e\}\in M^{*}(G)_{n-k}$, a contradiction. Otherwise, $H_{1}\in M^{*}(G)_{n-k}$, again a contradiction. This completes the proof of the theorem. $\square$

Proof of the following theorem follows from Lemma 2.1 and Theorem 2.3.

Theorem 2.4

Let G be a unicyclic graph with nullity $k>0$. Then the nonzero eigenvalues of G do not satisfy the (anti-) reciprocal eigenvalue property.

3 Noncorona Graphs with Strong Anti-Reciprocal Eigenvalue Property

In this section, we present some classes of noncorona graphs, namely ${{\mathcal {N}}}{{\mathcal {C}}}^{s}_{1}(s=1,2,3)$, ${{\mathcal {N}}}{{\mathcal {C}}}^{s}_{2}$ ($s=1,2$) and ${{\mathcal {N}}}{{\mathcal {C}}}^{s}_{3}$ ($i=1,2,3,4$), and prove that these classes of graphs satisfy property (-SR).

3.1 Some Lemmas

Ahmad et al. [1] gave the following necessary and sufficient condition for a polynomial of even degree to satisfy property (-SR).

Lemma 3.1

[1] A polynomial $P(x)=a_{0}x^{2n}+a_{1}x^{2n-1}+\cdots +a_{2n}$ satisfies the property (-SR) if and only if

$$\begin{aligned} a_{2n-i}=\left\{ \begin{array}{ll} a_{i}&{}\quad \text {if}\quad i\,\text {and}\,n\,\text {have the same parity,}\\ -a_{i}&{}\quad \text {otherwise} \end{array}\right. \end{aligned}$$

for all $i=0,1,2,\ldots ,2n$.

Remark 3.2

Let $P_{1}(x)=\displaystyle \sum _{i=0}^{2n}a_{2n-i}x^{2n-i}$ and $P_{2}(x)=\displaystyle \sum _{i=0}^{2n-2k}b_{2n-2k-i}x^{2n-2k-i}$ be two polynomials of degree 2n and $2n-2k$, respectively. Let $P(x)=\sum _{i=0}^{2n}c_{2n-i}x^{2n-i}=P_{1}(x)+cx^{k}P_{2}(x)$, where c is any constant. Then

$$\begin{aligned} c_{2n-i}=\left\{ \begin{array}{ll} a_{2n-i}+c\,b_{2n-(i+k)}&{} \quad \text {if}\quad i=k,k+1,\ldots ,2n-k,\\ a_{2n-i}&{}\quad \text {otherwise.} \end{array}\right. \end{aligned}$$

If both $P_{1}(x)$ and $P_{2}(x)$ satisfy the property (-SR), then

$$\begin{aligned}(-1)^{n}c_{2n-i}&=\left\{ \begin{array}{ll} (-1)^{n}a_{2n-i}+c(-1)^{n}\,b_{2n-(i+k)}&{}\quad \text {if}\quad i=k,k+1,\ldots ,2n-k,\\ (-1)^{n}a_{2n-i}&{}\quad \text {otherwise.} \end{array}\right. \\&=\left\{ \begin{array}{ll} (-1)^{i}(a_{i}+c\,b_{i-k})&{}\quad \text {if}\quad i=k,k+1,\ldots ,2n-k,\\ (-1)^{i}a_{i}&{}\quad \text {otherwise.} \end{array}\right. \\&=(-1)^{i}c_{i}. \end{aligned}$$

Thus, P(x) satisfies property (-SR).

The following is the well-known Schur complement formula.

Lemma 3.3

[3] Let $A=\left[ \begin{array}{cc}P&{}Q\\ R&{}S\end{array}\right]$ be a block matrix. Let P and S be square matrices.

• a. If P is invertible, then $\det (A)=\det (P)\det (S-RP^{-1}Q)$.

• b. If S is invertible, then $\det (A)=\det (S)\det (P-QS^{-1}R)$.

The inverse of a partitioned matrix can be computed using the following lemma.

Lemma 3.4

[3] Let $A=\left[ \begin{array}{cc}P&{}Q\\ R&{}S\end{array}\right]$ be a block matrix. Let P be an invertible square matrix and S be a square matrix. Then A is invertible if and only if the Schur complement of P, i.e., $P_{c}:=S-RP^{-1}Q$ is invertible. Moreover,

$$\begin{aligned} A^{-1}= \left[ \begin{array}{cc}P^{-1}+P^{-1}QP_{c}^{-1}RP^{-1}&{}-P^{-1}QP_{c}^{-1}\\ -P_{c}^{-1}RP^{-1}&{}P_{c}^{-1}\end{array}\right] . \end{aligned}$$

In Barik et al. [5] obtained the following lemma.

Lemma 3.5

[5] Let G be an r-regular graph of order m and $H = G\circ K_1$. Then

${\textbf {1}}^{T}(xI_{2m}-A(H))^{-1}{} {\textbf {1}}=\dfrac{(2x-r+2)m}{x^2-rx-1}.$

Lemma 3.6

[5] Let A be a square matrix of order n and $A[1,2,\ldots ,\ell ]$ be the square matrix obtained by removing the 1st, 2nd, $\ldots$, $\ell$-th rows and columns of the matrix A. Then for $1\le k\le n$ and for constants $m_{1},\,m_{2},\ldots ,\,m_{k}$, we have

$$\begin{aligned} \det \left( A+\sum _{i=1}^{k}\,m_{i}\,E_{ii}\right)&=\det (A)+\sum _{i=1}^{k}\,m_{i}\,\det (A[i])+\sum _{\underset{i<j}{i,j=1}}^{k}\,m_{i}\,m_{j}\,\det (A[i,j])\\&\quad +\cdots +m_{1}\,m_{2}\cdots m_{k}\,\det (A[1,2,\ldots ,k]). \end{aligned}$$

Let G and H be two graphs with vertex sets $V(G)=\{v_{1},\,v_{2},\ldots ,\,v_{n}\}$ and $V(H)=\{u_1,\,u_2,\ldots ,\,u_m\}$, respectively. The Kronecker product [19] of G and H, denoted by $G\otimes H$, is the graph with vertex set $V(G)\times V(H)$, and two vertices $(v_{i},\,u_{k})$ and $(v_{j},\,u_{\ell })$ are adjacent if and only if $v_{i}$ and $v_{j}$ are adjacent in G and also, $u_{k}$ and $u_{\ell }$ are adjacent in H. For example, if $V(K_{2})=\{v_{1},v_{2}\}$, $V(C_{m})=\{u_{1},u_{2},\ldots ,u_{m}\}$ and $E(C_{m})=\{u_{i}u_{i+1}| 1\le i\le m-1 \}\cup \{u_{m}u_{1}\}$, then $G=K_{2}\otimes C_{m}$ is the graph with vertex set $V(G)=\{(v_{1},u_{1}),(v_{1},u_{2}),\ldots ,(v_{1},u_{m}),(v_{2},u_{1}), (v_{2},u_{2}),\ldots ,(v_{2},u_{m})\}$ and edge set $E(G)=\{(v_{1},u_{i})(v_{2},u_{i+1}),(v_{1},u_{i+1})(v_{2},u_{i})| 1\le i\le m-1\}\cup \{(v_{1},u_{m})(v_{2},u_{1})\}\cup \{(v_{1},u_{1})(v_{2},u_{m})\}$. Thus, if m is odd, $K_{2}\otimes C_{m}$ is isomorphic to $C_{2m}$ and if m is even, then $K_{2}\otimes C_{m}$ is isomorphic to disjoint union of two cycles $C_{m}$.

The following lemma is used to prove our main results.

Lemma 3.7

Let C be a (0,1)-square matrix of order m such that $C{\textbf {1}}_{m}=C^{T}{} {\textbf {1}}_{m}=r{\textbf {1}}_{m}$. Let H be the bipartite graph with adjacency matrix $A(H)=\left[ \begin{array}{cc} 0_{m}&{}C\\ C^{T}&{}0_{m} \end{array}\right]$ and let $G_{1}=H\circ K_{1}$. If $E_{m}=\left[ \begin{array}{c}\mathrm{1}_{m}\\ \mathrm{0}_{m}\\ \mathrm{0}_{m}\\ \mathrm{1}_{m}\end{array}\right]$. Then

$$\begin{aligned} E_{m}^{T}(xI-A(G_{1}))^{-1}E_{m}=\dfrac{(2x^2-r^{2}+2r-2)xm}{x^{4}-(r^2+2)x^2+1}. \end{aligned}$$

Proof

By proper vertex labeling of the graph $G_{1}$, we obtain

$$\begin{aligned} xI-A(G_{1})=\left[ \begin{array}{cc|cc}xI_{m}&{}0_{m}&{}-I_{m}&{}0_{m}\\ 0_{m}&{}xI_{m}&{}0_{m}&{}-I_{m}\\ \hline -I_{m}&{}0_{m}&{}xI_{m}&{}-C\\ 0_{m}&{}-I_{m}&{}-C^{T}&{}xI_{m} \end{array}\right] . \end{aligned}$$

Applying Lemma 3.4 to the above partition matrix, we get

$$\begin{aligned} \Big (xI-A(G_{1})\Big )^{-1}=\left[ \begin{array}{cc} \dfrac{1}{x}I_{2m}+\dfrac{1}{x^2}M_{2}^{-1}&{}\dfrac{1}{x}M_{2}^{-1}\\ \dfrac{1}{x}M_{2}^{-1}&{}M_{2}^{-1} \end{array}\right] , \end{aligned}$$

(3.1)

where

$$\begin{aligned} M_{2}=\left[ \begin{array}{cc} \left( x-\dfrac{1}{x}\right) I_{m}&{}-C\\ -C^{T}&{}\left( x-\dfrac{1}{x}\right) I_{m} \end{array}\right] . \end{aligned}$$

By Lemma 3.4, the inverse of the matrix $M_{2}$ is

$$\begin{aligned} M_{2}^{-1}=\left[ \begin{array}{cc} \dfrac{x}{x^{2}-1}I_{m}+\dfrac{x^{2}}{(x^{2}-1)^2}CM_{1}^{-1}C^{T}&{} \dfrac{x}{x^{2}-1}CM_{1}^{-1}\\ \dfrac{x}{x^{2}-1}M_{1}^{-1}C^{T}&{}M_{1}^{-1} \end{array}\right] ,\end{aligned}$$

(3.2)

where $M_{1}=\left( x-\dfrac{1}{x}\right) I_{m}-\dfrac{x}{x^{2}-1}C^{T}C$.

Also, since $C{\textbf {1}}_{m}=C^{T}{} {\textbf {1}}_{m}=r{\textbf {1}}_{m}$, it is easy to see that

$$\begin{aligned} \mathbf{1}^{T}_{m}M_{1}^{-1}\mathbf{1}_{m}=\dfrac{mx(x^2-1)}{x^4-(r^2+2)x^2+1}. \end{aligned}$$

(3.3)

Now, employing Eq. (3.2) in (3.1), and then using (3.3), we get the required result. $\square$

3.2 Classes ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}, {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}, {{\mathcal {N}}}{{\mathcal {C}}}^{3}_{1}$

Let G be any graph of order n with vertex set $\{1,2,\ldots ,n\}$ and $G_{1}=G\circ K_{1}$. Let the pendent of $G_{1}$ adjacent with the vertex i of G be denoted by $i^{\prime }$.

Fig. 1

Graph $\Gamma _1=G[C_{4},n]\in {{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}$

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}$:

Let $\{v_{1}, v_{2}, \ldots , v_{m}\}$ and $\{u_{1}, u_{2}, \ldots , u_{m}\}$ be two disjoint vertex independent sets of the cycle $C_{2m}$. The graph $\Gamma =G[C_{2m},n]\in {{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}$ is obtained by taking n copies of corona graph $C_{2m}\circ K_{1}$ and joining the vertex $v_{i}$ $(1\le i\le m)$ of each copy of $C_{2m}\circ K_{1}$ to a new vertex x, say, and also by joining all the pendent vertices adjacent with the vertices $u_{1},\,u_{2},\ldots ,\,u_{m}$ (each copy of $C_{2m}\circ K_{1}$) to the vertex x, and then joining the vertex x with a new vertex, say, y (see Fig. 1).

In the following example we define a graph $\Gamma _{1}\in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{1}$ and list all its eigenvalues which indeed satisfies property (-SR).

Example 3.8

Let $\Gamma _1=G[C_{4},2] \in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{1}$. Then the spectrum of $\Gamma _1$ consists of $\pm 1$ with multiplicity 5; $1\pm \sqrt{2}$; $-1\pm \sqrt{2}$; $2\pm \sqrt{3}$ and $-2\pm \sqrt{3}$. Therefore, the graph $\Gamma _1$ satisfies property (-SR).

In the following theorem, we prove that the graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{1}$ satisfies the property (-SR).

Theorem 3.9

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}$ satisfies the property (-SR).

Proof

By proper vertex labeling of the graph $\Gamma$, we have the adjacency matrix of $\Gamma$ as:

$$\begin{aligned} A(\Gamma )=\left[ \begin{array}{cccc} I_{n}\otimes \left[ \begin{array}{cc} {{\textbf {0}}}_{2m\times 2m}&{}I_{2m}\\ I_{2m}&{}\begin{array}{cc} {{\textbf {0}}}_{m\times m}&{}C\\ C^{T}&{}{{\textbf {0}}}_{m\times m} \end{array} \end{array}\right] &{}{\textbf {1}}_{n}\otimes \left[ \begin{array}{c}{} \mathbf{{1}}_{m}\\ \mathbf{{0}}_{m}\\ \mathbf{{0}}_{m}\\ \mathbf{{1}}_{m}\\ \end{array}\right] &{}{{\textbf {0}}}_{4mn\times 1}\\ {\textbf {1}}^{T}_{n}\otimes \left[ \begin{array}{c}{} {\textbf {1}}_{m}\\ {{\textbf {0}}}_{m}\\ {{\textbf {0}}}_{m}\\ {\textbf {1}}_{m}\\ \end{array}\right] ^{T}&{}0&{} 1\\ \\ {{\textbf {0}}}_{1\times 4mn}&{} 1&{} 0 \end{array}\right] , \end{aligned}$$

where $\left[ \begin{array}{cc} {{\textbf {0}}}_{m\times m}&{}C\\ C^{T}&{}{{\textbf {0}}}_{m\times m} \end{array}\right]$ is the adjacency matrix of cycle $C_{2m}$.

Let $v_{i}$ and $w_{i}$ be the singular vector pairs corresponding to the singular values $\lambda _{i}$ of C for $i=1,2,\ldots ,m$. Without loss of generality we can assume that the sets $\{v_{1},v_{2},\ldots , v_{m}\}$ and $\{w_{1},w_{2},\ldots , w_{m}\}$ are orthonormal. Also, since $C{\textbf {1}}_{m}=2{\textbf {1}}_{m}$ and $C^{T}1_{m}=2{\textbf {1}}_{m}$, we can further assume that $v_{1}=w_{1}=\frac{1}{\sqrt{m}}{} {\textbf {1}}_{m}$.

Let $Z^{+}_{i}{:=}\left[ \begin{array}{c} v_{i}\\ w_{i}\\ \delta _{i}v_{i}\\ \delta _{i}w_{i} \end{array}\right]$ and $Z^{-}_{i}=\left[ \begin{array}{c} v_{i}\\ {}-w_{i}\\ \delta ^{\prime }_{i}v_{i}\\ {}-\delta ^{\prime }_{i}w_{i} \end{array}\right] ,$ where $\delta _{i}$ and $\delta ^{\prime }_{i}$ are some nonzero scalars $(2\le i\le m)$. Then for $i=2,3,\ldots ,m$ and $j=1,2,\ldots ,n$, we obtain

$$\begin{aligned} A(\Gamma )\left[ \begin{array}{c}e_{j}\otimes Z_{i}^{+}\\ 0\\ 0\end{array}\right] =\left[ \begin{array}{c}e_{j}\otimes \delta _{i}\left[ \begin{array}{c} v_{i}\\ w_{i}\\ \frac{1+\lambda _{i}\delta _{i}}{\delta _{i}}v_{i}\\ \frac{1+\lambda _{i}\delta _{i}}{\delta _{i}}w_{i} \end{array}\right] \\ 0\\ 0\end{array}\right] . \end{aligned}$$

Therefore, $\left[ \begin{array}{c}e_{j}\otimes Z_{i}^{+}\\ 0\\ 0\end{array}\right]$ is an eigenvector corresponding to an eigenvalue $\delta _i$ of $A(\Gamma )$ if and only if $\frac{1+\lambda _i\delta _i}{\delta _i}=\delta _i$, that is, if and only if $\delta _{i}=\dfrac{\lambda _{i}\pm \sqrt{\lambda _{i}^2+4}}{2}.$ Thus, $\dfrac{\lambda _{i}\pm \sqrt{\lambda _{i}^2+4}}{2}$ is an eigenvalue corresponding to the eigenvector $\left[ \begin{array}{c}e_{j}\otimes Z_{i}^{+}\\ 0\\ 0\end{array}\right]$ of $A(\Gamma )$ with $\delta _{i}=\dfrac{\lambda _{i}\pm \sqrt{\lambda _{i}^2+4}}{2}.$ Similarly, it can be seen that $\dfrac{-\lambda _{i}\pm \sqrt{\lambda _{i}^2+4}}{2}$ is an eigenvalue corresponding to the eigenvector $\left[ \begin{array}{c}e_{j}\otimes Z_{i}^{-}\\ 0\\ 0\end{array}\right]$ of $A(\Gamma )$ with $\delta ^{\prime }_{i}=\dfrac{-\lambda _{i}\pm \sqrt{\lambda _{i}^2+4}}{2}.$

Now, let $Z^{+}_{1}=\left[ \begin{array}{c} {\textbf {1}}_{m}\\ {\textbf {1}}_{m}\\ \delta _{1}{} {\textbf {1}}_{m}\\ \delta _{1}{} {\textbf {1}}_{m} \end{array}\right]$ and $Z^{-}_{1} =\left[ \begin{array}{c} {\textbf {1}}_{m}\\ -{\textbf {1}}_{m}\\ \delta ^{\prime }_{1}{} {\textbf {1}}_{m}\\ -\delta ^{\prime }_{1}{} {\textbf {1}}_{m} \end{array}\right]$, where $\delta _{1}$ and $\delta ^{\prime }_{1}$ are some nonzero scalars.

Then for $j=2,3,\ldots ,n$, we have

$A(\Gamma )\left[ \begin{array}{c} (e_{1}-e_{j})\otimes Z^{+}_{1}\\ 0\\ 0 \end{array}\right] =\delta _{1}\left[ \begin{array}{c} (e_{1}-e_{j})\otimes Z^{+}_{1}\\ 0\\ 0 \end{array}\right]$ if and only if $\frac{1+2\delta _1}{\delta _1}=\delta _1$, that is, if and only if $\delta _{1}=1\pm \sqrt{2}$ and $A(\Gamma )\left[ \begin{array}{c} (e_{1}-e_{j})\otimes Z^{-}_{1}\\ 0\\ 0 \end{array}\right] =\delta ^{\prime }_{1}\left[ \begin{array}{c} (e_{1}-e_{j})\otimes Z^{-}_{1}\\ 0\\ 0 \end{array}\right]$ if and only if $\delta ^{\prime }_{1}=-1\pm \sqrt{2}$. Thus, $1\pm \sqrt{2}$ and $-1\pm \sqrt{2}$ are the eigenvalues of $A(\Gamma )$ with multiplicity $n-1$, respectively. Thus, we have listed $4mn-4$ eigenvalues of $A(\Gamma )$ corresponding to the $4mn-4$ orthogonal eigenvectors as described above. Let $t_{i}\,(i=1,\,2,\ldots ,\,6)$ be the remaining eigenvalues of $A(\Gamma )$ corresponding to the eigenvector $X_{i}$. Let $Y_{i}=\left[ \begin{array}{c}{} {\textbf {1}}_{n}\otimes (e^{i}_{4}\otimes {\textbf {1}}_{m})\\ 0\\ 0\end{array}\right]$, where $i=1,\,2,\,3,\,4$; $Y_{5}=\left[ \begin{array}{c} {{\textbf {0}}}_{4m}\\ 1\\ 0 \end{array}\right]$ and $Y_{6}=\left[ \begin{array}{c} {{\textbf {0}}}_{4m}\\ 0\\ 1 \end{array}\right]$. Then $Y_{i}$ ($i=1,\,2,\ldots ,\,6$) along with the above-described $4mn-4$ orthogonal eigenvectors of $A(\Gamma )$ form a set of $4mn+2$ linearly independent vectors of size $4mn+2$. Since $X_{i}$ is an eigenvector corresponding to the eigenvalue $t_{i}$, then $X_{i}=\displaystyle \sum _{i=1}^{6}a_{i}Y_{i}$ for some scalar $a_{i}$ (not all zero). Therefore, the equation $A(\Gamma )X_{i}=t_{i}X_{i}$ holds if and only if $a_{5}-a_{6}t_{i}=0; a_{4}-a_{2}t_{i}=0; a_{1}-a_{3}t_{i}+2a_{4}=0; a_{1}t_{i}-a_{3}-a_{5}=0; a_{2}+2a_{3}+a_{5}-a_{4}t_{i}=0; nma_{1}+a_{4}nm+a_{6}-a_{5}t_{i}=0.$ Equivalently, the equation $A(\Gamma )X_{i}=t_{i}X_{i}$ holds if and only if

$$\begin{aligned} \det \left[ \begin{array}{cccccc} -t_{i}&{}0&{}1&{}0&{}1&{}0\\ {}0&{}-t_{i}&{}0 &{}1&{}0&{}0\\ {}1&{}0&{}-t_{i}&{}2&{}0&{}0\\ {}0&{}1&{}2&{}-t_{i}&{}1 &{}0\\ {}nm&{}0&{}0&{}nm&{}-t_{i}&{}1\\ {}0&{}0&{}0&{}0&{}1&{} -t_{i}\end{array} \right] =0. \end{aligned}$$

Thus, the remaining six eigenvalues of $A(\Gamma )$ are the roots of the polynomial $(t^2-1)\,\Big (t^4-2(mn+3)t^2+1\Big )$. Hence, the eigenvalues of $\Gamma$ are $\dfrac{\lambda _{i}\pm \sqrt{\lambda ^2_{i}+4}}{2}$ with multiplicity $n(m-1)$; $\dfrac{-\lambda _{i}\pm \sqrt{\lambda ^2_{i}+4}}{2}$ with multiplicity $n(m-1)$; $1\pm \sqrt{2}$ with multiplicity $n-1$; $-1\pm \sqrt{2}$ with multiplicity $n-1$; ± 1; and the roots of the polynomial $p(t):=t^4-2(mn+3)t^2+1$. Since p(t) satisfies the property (-SR) (see Lemma 3.1) and the remaining eigenvalues of $\Gamma$ also satisfies the property (-SR), the theorem follows. $\square$

Remark 3.10

A natural question here to ask is that “Whether Theorem 3.9 is true for any r-regular bipartite graph other than the cycle $C_{2m}$?”. The answer is no. By replacing the cycle $C_{2m}$ by an r-regular bipartite graph ($r>2$), the class ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{1}$ cannot be generalized further to satisfy property (-SR), because the spectrum of the generalized graph will contain the four roots of the polynomial $t^6-(2mn+r^2+3)t^4+(mnr^2-2mnr+2mn+r^2+3)t^2-1$, which indeed satisfies property (-SR) only for $r=2$.

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$:

The graph $\Gamma =G[G_{1}, H_{1},H_{2},\ldots , H_{n}]\in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$ is obtained from the graphs $G_{1}$ and $H_{1}, H_{2},\ldots , H_{n}$, by joining the vertex i $(i=1,2,\ldots ,n)$ of G with the vertices $v_{it}$ and $u^{\prime }_{it}$ of $H_{i}$ for $t=1,2,\ldots ,p_{i}$ (for example, see Fig. 2).

In the following example we define a graph $\Gamma _{2}\in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{2}$ and list all its eigenvalues which indeed satisfies property (-SR).

Example 3.11

Let $\Gamma _2=G[G_{1}, H_{1}, H_{2}, H_{3}]\in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$ with $G_{1}=K_{3}\circ K_{1}$ and $H_{i}=C_{6}\circ K_{1}$ for $i=1,2$, and $H_{3}=C_{4}\circ K_{1}$. Then the spectrum of $\Gamma _2$ consists of $-3.78$; $-3.58$; $-2.95$; $-1.61$ with multiplicity 4; $-1.25$; $-1.21$; $-1$; $-1$; $-0.68$; $-0.61$ with multiplicity 4; $-0.33$; $-0.31$; $-0.24$; 0.26; 0.27; 0.33; 0.61 with multiplicity 4; 0.8; 0.82; 1; 1; 1.46; 1.61 with multiplicity 4; 3.02; 3.22; 4.13. Thus, $\Gamma _2$ satisfies property (-SR).

Fig. 2

Graph $\Gamma _2=G[G_{1}, H_{1}, H_{2}, H_{3}]\in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$ with $G_{1}=K_{3}\circ K_{1}$ and $H_{i}=C_{6}\circ K_{1}$ for $i=1,2$, and $H_{3}=C_{4}\circ K_{1}$

Remark 3.12

Adjacency matrix of $\Gamma =G[G_{1}, H_{1},H_{2},\ldots , H_{n}]\in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$: Let $k=4(p_{1}+p_{2}+\cdots +p_{n})$. By proper vertex labeling of $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$, we have the adjacency matrix of $\Gamma$ as $A(\Gamma )=\left[ \begin{array}{ccc}B&{}E&{}{{\textbf {0}}}_{k\times n}\\ E^{T}&{}A(G)&{}I_{n}\\ {{{\textbf {0}}}}_{n\times k}&{}I_{n}&{}{{{\textbf {0}}}}_{n\times n}\end{array}\right]$, where B is the square matrix of order k and E is the matrix of order $k\times n$, and are defined as

$$\begin{aligned} B=\left[ \begin{array}{ccccc} A(H_{1})&{}{{{\textbf {0}}}}&{}{{{\textbf {0}}}}&{}\ldots &{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}A(H_{2})&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots &{}\vdots \\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}A(H_{n-1})&{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}0&{}A(H_{n}) \end{array}\right] ,\quad E=\left[ \begin{array}{ccccc} E_{1}&{}{{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}E_{2}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots &{}\vdots \\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}E_{n-1}&{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}&{}E_{n} \end{array}\right] , \end{aligned}$$

where $E_{i}=\left[ \begin{array}{c}{{\textbf {1}}}_{p_{i}}\\ {{\textbf {0}}}_{p_i}\\ {{\textbf {0}}}_{p_{i}}\\ {{\textbf {1}}}_{p_{i}}\\ \end{array}\right] .$

We now show that the graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$ satisfies the property (-SR).

Theorem 3.13

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{1}$ satisfies the property (-SR).

Proof

By Remark 3.12, we get

$$\begin{aligned} \det \Big (xI-A(\Gamma )\Big )=\det \left( \begin{array}{ccc}xI_{k}-B&{}-E&{}{{\textbf {0}}}_{k\times n}\\ -E^{T}&{}xI_{n}-A(G)&{}-I_{n}\\ {{\textbf {0}}}_{n\times k}&{}-I_{n}&{}xI_{n}\end{array}\right) . \end{aligned}$$

Applying Lemma 3.3 to the above determinant, we obtain

$$\begin{aligned}&P(\Gamma ;x)\\&\quad =\det (xI_{k}-B)~ \det \left( \left[ \begin{array}{cc} xI_{n}-A(G)&{}-I_{n}\\ -I_{n}&{}xI_{n} \end{array}\right] -\left[ \begin{array}{c}-E^{T}\\ {{\textbf {0}}}_{n\times k}\end{array}\right] (xI_{k}-B)^{-1}\right. \left. \left[ \begin{array}{c}-E^{T}\\ {{\textbf {0}}}_{n\times k}\end{array}\right] ^{T}\right) \\&\quad =\prod _{i=1}^{n}P(H_{i};x)\\&\qquad \times \det \left( \left[ \begin{array}{cc} xI_{n}-A(G)&{}-I_{n}\\ -I_{n}&{}xI_{n} \end{array}\right] -\left[ \begin{array}{cc} \displaystyle \sum _{i=1}^{n}(E^{T}_{i}(xI_{4p_{i}}-A(H_{i}))^{-1}E_{i})E_{ii}&{}{\textbf {0}}_{n\times n}\\ {{\textbf {0}}}_{n\times n}&{}{{\textbf {0}}}_{n\times n} \end{array}\right] \right) . \end{aligned}$$

By Lemmas 3.6 and 3.7, we obtain

$$\begin{aligned}&P(\Gamma ;x)\\&\quad =\prod _{i=1}^{n}P(H_{i};x)\times \det \left( \left[ \begin{array}{cc} xI_{n}-A(G)&{}-I_{n}\\ -I_{n}&{}xI_{n} \end{array}\right] -\left[ \begin{array}{cc} \displaystyle \sum _{i=1}^{n}\dfrac{2p_{i}x(x^2-1)}{x^4-6x^2+1}E_{ii}&{}{{\textbf {0}}}_{n\times n}\\ {{\textbf {0}}}_{n\times n}&{}{{\textbf {0}}}_{n\times n} \end{array}\right] \right) \\&\quad =\prod _{i=1}^{n}P(H_{i};x)\times \Bigg [\det \Big (xI_{2n}-A(G_{1})\Big )\\&\qquad +\dfrac{2x(1-x^2)}{x^4-6x^2+1}\sum _{i=1}^{n}p_{i}\det \Big (xI_{2n}-A(G_{1})[i]\Big )\\&\qquad +\left( \dfrac{2x(1-x^2)}{x^4-6x^2+1}\right) ^2\sum _{\underset{i<j}{i,j=1}}^{n}p_{i}p_{j}\det \Big (xI_{2n}-A(G_{1})[i,j]\Big )\\&\qquad +\cdots +\left( \dfrac{2x(1-x^2)}{x^4-6x^2+1}\right) ^np_{1}p_{2}\ldots p_{n}\det \Big (xI_{2n}-A(G_{1})[1,2,3,\ldots ,n]\Big )\Bigg ]. \end{aligned}$$

Thus,

$$\begin{aligned} P(\Gamma ;x)&=\prod _{i=1}^{n}\dfrac{P(H_{i};x)}{x^4-6x^2+1}\times \Bigg [(x^4-6x^2+1)^nP(G_{1},x)\\&\quad +2x^2(1-x^2)(x^4-6x^2+1)^{n-1}\sum _{i=1}^{n}p_{i}P(G_{1}\backslash \{i,i^{\prime }\};x) \\&\quad +4x^4(1-x^2)^2(x^4-6x^2+1)^{n-2}\sum _{\underset{i<j}{i,j=1}}^{n}p_{i}p_{j}P(G_{1}\backslash \{i,i^{\prime },j,j^{\prime }\};x)\\&\quad +\cdots +2^nx^{2n}(1-x^2)^np_{1}p_{2}\ldots p_{n}\Bigg ]. \end{aligned}$$

Note that for $j=1,2,\ldots ,n-1$ and $i_{\ell }\in \{1,2,\ldots ,n\}$, $G_{1}\backslash \{i_1,i_2,\ldots ,i_j,i^{\prime }_1,i^{\prime }_2,\ldots , i^{\prime }_j\}$ is a corona graph and hence $P\Big (G_{1}\backslash \{i_1,i_2,\ldots ,i_j,i^{\prime }_1,i^{\prime }_2,\ldots , i^{\prime }_j\};x\Big )$ satisfies the property (-SR). Since

$$\begin{aligned}&\dfrac{P(H_{i};x)}{x^4-6x^2+1}, \text{ and } ~(1-x^2)^i(x^4-6x^2+1)^{n-i}P\Big (G_{1}\backslash \{i_1,i_2,\ldots , i_j,i^{\prime }_1,i^{\prime }_2,\ldots , \\&i^{\prime }_j\};x\Big ) \end{aligned}$$

satisfies the property (-SR), the theorem follows by Remark 3.2. $\square$

Remark 3.14

Theorem 3.13 can be generalized further by taking $n_{i}\ge 1$ copies of $H_{i}$ corresponding to each vertex i of G. Thus, Theorem 3.13 is more general result than Theorem 3.9. However, Theorem 3.9 helps us to observe that the bipartite regular graphs cannot be used instead of cycle $C_{2p_{i}}$ in our constructions.

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{3}_{1}$:

The graph $\Gamma =G[G_{1},H_{1},H_{2},\ldots ,H_{k},F_{k+1},F_{k+2},\ldots ,F_{\ell }]~(k+\ell \le n)\in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{3}$ is a graph obtained from the graphs $G_{1}$, $H_{i}$ and $F_{j}$, by joining the vertex i $(i=1,2,\ldots ,n)$ of G with the vertices $v_{it}$ and $u^{\prime }_{it}$ ($t=1,2,\ldots , p_{i}$) of $H_{i}$ for $i=1,2,\ldots ,k$, and then joining with all the vertices of $F_{i}$ for $i=k+1,k+2,\ldots ,\ell$.

Note that the graph $\Gamma _1\in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{3}$ is a generalization of the graph $\Gamma _2\in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{2}$. Similar to the proof of Theorem 3.13, we obtain the following theorem.

Theorem 3.15

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{1}^{3}$ satisfies property (-SR).

3.3 Classes ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{2}, {{\mathcal {N}}}{{\mathcal {C}}}^{2}_{2}$

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{2}$:

For any r-regular graph G of order n, let $G_{1}=G\circ K_{1}$. The graph $\Gamma =G[G_1,H_{1}, H_{2}, \ldots , H_{k}, F_{1}, F_{2},\ldots , F_{\ell }]\in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ is obtained from the graphs $G_{1}$, $H_{i}$ ($i=1,2,\ldots , k$) and $F_{j}$ ($j=1,2,\ldots ,\ell$), by joining all quasi pendent vertices of $G_{1}$ with the vertices of $F_{j}$ for $j=1,2,\ldots ,\ell$, and then joining with the vertices $v_{it}$ and $u^{\prime }_{it}$ ($t=1,2,\ldots ,p_{i}$) of $H_{i}$ for $i=1,2,\ldots ,k$ (for example, see Fig. 3).

In the following example we define a graph $\Gamma _{3}\in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ and list all its eigenvalues which indeed satisfies property (-SR).

Example 3.16

Let $\Gamma _3=G[G_{1}, H_{1}, F_{1}]\in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ with $G_{1}=K_{2}\circ K_{1}$, $H_{1}=C_{6}\circ K_{1}$ and $F_{1}=C_{4}\circ K_{1}$. Then the spectrum of $\Gamma _3$ consists of $-4.82$; $-2.41$; $-1.61$ with multiplicity 3; $-1.56$; $-1$; $-1$; $-0.61$; $-0.61$; $-0.41$; $-0.14$; 0.20; 0.41; 0.61 with multiplicity 3; 0.639; 1; 1; 1.61; 1.61; 2.41; 6.69. Thus, $\Gamma _3$ satisfies property (-SR).

Fig. 3

Graph $\Gamma _3=G[G_{1}, H_{1}, F_{1}]\in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ with $G_{1}=K_{2}\circ K_{1}$, $H_{1}=C_{6}\circ K_{1}$ and $F_{1}=C_{4}\circ K_{1}$

Remark 3.17

Let $k_{1}=4(p_{1}+p_{2}+\cdots +p_{k})$ and $k_{2}=2(m_{1}+m_{2}+\cdots +m_{\ell })$. By proper vertex labeling of the graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1},$ its adjacency matrix can be written as:

$$\begin{aligned} \left[ \begin{array}{cccc} H&{}{\mathrm {0}}_{k_{1},k_{2}}&{}E&{}{{\textbf {0}}}_{k_{1},n}\\ {{\textbf {0}}}_{k_{2},k_{1}}&{}F&{}J_{k_{2},n}&{}{{\textbf {0}}}_{k_{2},n}\\ E^{T}&{}J_{n,k_{2}}&{}A(G)&{}I_{n}\\ {{\textbf {0}}}_{n,k_{1}}&{}{{\textbf {0}}}_{n,k_{2}}&{}I_{n}&{}{{\textbf {0}}}_{n\times n} \end{array}\right] , \end{aligned}$$

where H is a square matrix of order $k_{1}$ and E is the rectangular matrix of order $k_{1}\times n$, which are defined as:

$$\begin{aligned} H=\left[ \begin{array}{ccccc} A(H_{1})&{}{{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}A(H_{2})&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots &{}\vdots \\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}A(H_{k-1})&{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}&{}A(H_{k}) \end{array}\right] \quad \hbox {and} \quad E=\left[ \begin{array}{ccccc} E_{1}&{}E_{1}&{}E_{1}&{}\ldots &{}E_{1}\\ E_{2}&{}E_{2}&{}E_{2}&{}\ldots &{}E_{2}\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ E_{k-1}&{}E_{k-1}&{}\ldots &{}E_{k-1}&{}E_{k-1}\\ E_{k}&{}E_{k}&{}\ldots &{}E_{k}&{}E_{k} \end{array}\right] , \end{aligned}$$

$E_{i}=\left[ \begin{array}{c} {\textbf {1}}_{p_{i}}\\ {{\textbf {0}}}_{p_{i}}\\ {{\textbf {0}}}_{p_{i}}\\ {{\textbf {1}}}_{p_{i}} \end{array}\right] .$ Also, F is a square matrix of order $k_{2}$ given by

$$\begin{aligned} F=\left[ \begin{array}{ccccc} A(F_{1})&{}{{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}A(F_{2})&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}\\ \vdots &{}\vdots &{}\ddots &{}\vdots &{}\vdots \\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}A(F_{\ell -1})&{}{{\textbf {0}}}\\ {{\textbf {0}}}&{}{{\textbf {0}}}&{}\ldots &{}{{\textbf {0}}}&{}A(F_{\ell }) \end{array}\right] . \end{aligned}$$

Theorem 3.18

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ satisfies the property (-SR).

Proof

By Remark 3.17, we get

$$\begin{aligned} \det \Big (xI-A(\Gamma )\Big )=\det \left( \begin{array}{cccc} xI_{k_{1}}-H&{}{{\textbf {0}}}_{k_{1},k_{2}}&{}-E&{}{{\textbf {0}}}_{k_{1},n}\\ {{\textbf {0}}}_{k_{2},k_{1}}&{}xI_{k_{2}}-F&{}-J_{k_{2},n}&{}{{\textbf {0}}}_{k_{2},n}\\ -E^{T}&{}-J_{n,k_{2}}&{}xI_{n}-A(G)&{}-I_{n}\\ {{\textbf {0}}}_{n,k_{1}}&{}{{\textbf {0}}}_{n,k_{2}}&{}-I_{n}&{}xI_{n} \end{array}\right) . \end{aligned}$$

Applying Lemma 3.3 to the above determinant, we get

$$\begin{aligned}&\det \Big (xI-A(\Gamma )\Big )\nonumber \\&\quad =\det \Big ((xI_{k_{1}}-H)(xI_{k_{2}}-F)\Big )\,\det \left( \left[ \begin{array}{cc} xI_{n}-A(G)&{}-I_{n}\\ -I_{n}&{}xI_{n} \end{array}\right] \right. \nonumber \\&\qquad \left. -\left[ \begin{array}{cc} E^T (xI_{k_{1}}-H)^{-1}E+J_{n\times k_{1}}(xI_{n_{1}}-F)^{-1}J_{k_{1}\times n}&{}{{\textbf {0}}}_{n\times n}\\ {{\textbf {0}}}_{n\times n}&{}{{\textbf {0}}}_{n\times n} \end{array}\right] \right) \nonumber \\&\quad =\det \Big ((xI_{k_{1}}-H)(xI_{k_{2}}-F)\Big )\times \det \left[ \begin{array}{cc} xI_{n}-(A(G)+\delta J_{n})&{}-I_{n}\\ -I_{n}&{}xI_{n} \end{array}\right] , \end{aligned}$$

(3.4)

where

$$\begin{aligned} \delta =\displaystyle \sum _{i=1}^{k}E^{T}_{i}\Big (xI_{4p_{i}}-A(H_{i})\Big )^{-1}E_{i}+\displaystyle \sum _{j=1}^{\ell }{} {\textbf {1}}^{T}_{2m_{j}}\Big (xI_{2m_{i}}-A(F_{j})\Big )^{-1}{} {\textbf {1}}_{2m_{j}}. \end{aligned}$$

Now, by Lemmas 3.5 and 3.7, we obtain

$$\begin{aligned} \delta =\dfrac{k_{1}x(x^2-1)}{2(x^4-6x^2+1)}+\dfrac{k_{2}x}{x^{2}-2x-1}. \end{aligned}$$

Applying Lemma 3.3 to Eq. (3.4), we get

$$\begin{aligned} \det \Big (xI-A(\Gamma )\Big )&=\det \Big ((xI_{k_{1}}-H)(xI_{k_{2}}-F)\Big )\,x^{n}\det \Big (\Big (x-\dfrac{1}{x}\Big )I_{n}\nonumber \\&\quad -(A(G)+\delta J_{n})\Big ). \end{aligned}$$

(3.5)

Since G is an r-regular graph, it is easy to see that the matrices $\Big (x-\dfrac{1}{x}\Big )I_{n}-\Big (A(G)+\delta J_{n}\Big )$ and $\Big (x-\dfrac{1}{x}\Big )I_{n}-D$ are similar, where D is the diagonal matrix given by $D=diag(r+n\delta ,\lambda _{2},\lambda _{3},\ldots ,\lambda _{n})$. Thus, Eq. (3.5) reduces to

$$\begin{aligned} \det \Big (xI-A(\Gamma )\Big )&=\det \Big ((xI_{k_{1}}-H)(xI_{k_{2}}-F)\Big )R(x)\prod _{i=2}^{n}(x^{2}-\lambda _{i}x-1), \end{aligned}$$

where $R(x)=\dfrac{1}{2}(x^4-6x^2+1)^{-1}\Big [2x^6-2rx^5-(k_{1}n+2k_{2}n+14)x^4-(4k_{2}n-12r)x^3+(k_{1}n+2k_{2}n+14)x^2-2rx-2\Big ]$.

Therefore,

$$\begin{aligned} P(\Gamma ;x)=R(x)\displaystyle \prod _{i=1}^{k}P(H_{i};x)\displaystyle \prod _{j=1}^{\ell }P(F_{j};x)\displaystyle \prod _{i=2}^{n}(x^{2}-\lambda _{i}x-1). \end{aligned}$$

Since $H_{i}$ and $F_{j}$ are corona graphs, $P(H_{i};x)$ and $P(F_{j};x)$ satisfy the property (-SR). Also by Lemma 3.1, R(x) satisfies property (-SR). Thus, the graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{1}$ satisfies property (-SR). $\square$

The join of two graphs G and H, denoted by $G\vee H$, is the graph obtained by joining each vertex of G with every vertex of H. In literature, several variants of join graphs have been introduced and their spectral properties are studied, for example, see [11].

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{2}_{2}$:

Let G and K be two regular graphs. Let $G_{1}= (G\vee K)\circ K_{1}$. The graph $\Gamma =G[G_{1}, H_{1}, H_{2}, \ldots ,H_{k}, F_{1}, F_{2},\ldots ,F_{\ell }]\in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^2$ is obtained from the graphs $G_{1}$, $H_{i}$ ($i=1,2,\ldots ,k$), $F_{j}$ ($j=1,2,\ldots ,\ell$) by joining the vertices of G with the vertices $v_{it}$ and $u^{\prime }_{it}$ ($t=1,2,\ldots ,p_{i}$) of $H_{i}$ for $i=1,2,\ldots ,k$, and then joining the vertices of K with the vertices of $F_{j}$ for $j=1,2,\ldots ,\ell$.

The following theorem can be obtained similar to that of Theorem 3.18.

Theorem 3.19

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}_{2}^{2}$ satisfies property (-SR).

3.4 Classes ${{\mathcal {N}}}{{\mathcal {C}}}^{1}_{3}$, ${{\mathcal {N}}}{{\mathcal {C}}}^{2}_{3}$, ${{\mathcal {N}}}{{\mathcal {C}}}^{3}_{3}$, ${{\mathcal {N}}}{{\mathcal {C}}}^{4}_{3}$

The subdivision graph of G, denoted by S(G), is obtained by inserting a new vertex into every edge of G. The graph Q(G) is obtained from S(G) by adding an edge between two new vertices whenever the corresponding edges are adjacent. The graph R(G) is obtained by introducing a new vertex corresponding to every edge of G, and then joining the new vertex to the end vertices of the corresponding edge. The total graph of G, denoted by T(G), is obtained from R(G) by adding an edge between two new vertices whenever the corresponding edges are adjacent.

Class ${{\mathcal {N}}}{{\mathcal {C}}}^{s}_{3}$ ($s=1,2,3,4$):

Suppose that G is an r-regular graph of order n and size m. Let $K_{1}(G)=S(G)$, $K_{2}(G)=Q(G)$, $K_{3}(G)=R(G)$ and $K_{4}(G)=T(G)$. Also let $G_{s}=K_{s}(G)\circ K_{1}$. For $s=1,\,2,\,3$, the graph $\Gamma =G[G_{s},H_{1},H_{2},\ldots ,H_{k},F_{1},F_{2},\ldots ,F_{\ell }]\in {{\mathcal {N}}}{{\mathcal {C}}}^{s}_{3}$ is obtained from the graphs $G_{s}$, $H_{i}$ $(i=1,2,\ldots ,k)$ and $F_{j}$ ($j=1,2,\ldots ,\ell$) by joining the new vertices of $K_{s}(G)$ with the vertices $v_{it}$ and $u^{\prime }_{it}$ ($t=1,2,\ldots ,p_{i}$) of $H_{i}$ for $i=1,2,\ldots ,k$, and then joining the old vertices of $K_{s}(G)$ with the vertices of $F_{j}$ for $j=1,2,\ldots ,\ell$.

Fig. 4

Graph $\Gamma _4=G[G_{1}, H_{1}, F_{1}]\in {{\mathcal {N}}}{{\mathcal {C}}}_{3}^{1}$ with $G_{1}=K_{1}(C_{3})\circ K_{1}$, $H_{1}=C_{6}\circ K_{1}$ and $F_{1}=C_{4}\circ K_{1}$

In the following example we define a graph $\Gamma _{4}\in {{\mathcal {N}}}{{\mathcal {C}}}_{3}^{1}$ and list all its eigenvalues which indeed satisfies property (-SR).

Example 3.20

Let $\Gamma _4=G[G_{1}, H_{1}, F_{1}]\in {{\mathcal {N}}}{{\mathcal {C}}}_{3}^{1}$ with $G_{1}=K_{1}(C_{3})\circ K_{1}$, $H_{1}=C_{6}\circ K_{1}$ and $F_{1}=C_{4}\circ K_{1}$. Then the spectrum of $\Gamma _4$ consists of $-5.68$; $-3.73$; $-2.41$; $-1.61$ with multiplicity 4; $-1$; $-1$; $-0.97$; $-0.61$ with multiplicity 4; $-0.21$; $-0.15$; 0.17; 0.26; 0.41; 0.61 with multiplicity 4; 1; 1; 1.03; 1.61 with multiplicity 4; 4.62; 6.64. Thus, $\Gamma _4$ satisfies property (-SR).

Remark 3.21

The adjacency matrix of $K_{s}(G)$ is $\left[ \begin{array}{cc} A_s&{}B\\ B^{T}&{}C_s \end{array}\right]$, where $A_s$ is a square matrix of order m such that $A_{s}=0$ for $s=1,3$ and $A_{s}=A(L(G)),$ the adjacency matrix of the line graph of G for $s=2,4$; $C_{s}$ is a square matrix of order n such that $C_{s}=0$ for $s=1,2$ and $C_{s}=A(G)$ for $s=3,4$, and B is the incidence matrix of G.

Let $s^{\prime }=0$ for $s=1,3$ and $s^{\prime }=1$ for $s=2,4$. Also, let $s^{\prime \prime }=0$ for $s=1,2$ and $s^{\prime \prime }=1$ for $s=3,4$. Define $B^{\prime }$ to be the $m\times n$ rectangular diagonal matrix with diagonal entry $b_{ii}=\sqrt{\lambda _{i}+r_{i}}$ for $i=1,2,\ldots ,n$ and $b_{ii}=0$ for $i>n$. Since G is regular, from the singular value decomposition of the matrix B, we can see that the matrix $\left[ \begin{array}{cc} A_{s}+aJ_{m}&{}B\\ B^{T}&{}C_{s}+bJ_{n} \end{array}\right]$ is similar to the matrix $\left[ \begin{array}{cc} A_{s}^{\prime }&{}B^{\prime }\\ {B^{\prime }}^{T}&{}C^{\prime }_{s} \end{array}\right]$, where $A^{\prime }_{s}$ is the diagonal matrix of order m with diagonal entry $a_{11}=s^{\prime }(2r-2)+ma$, $a_{ii}=s^{\prime }(\lambda _{i}+r-2)$ for $2\le i\le n$ and $a_{ii}=-2s^{\prime }$ for $i>n$, and $C_{s}^{\prime }$ is a diagonal matrix of order n with its diagonal entry $c_{11}=s^{\prime \prime }r+bn$ and $c_{ii}=s^{\prime \prime }\lambda _{i}$ for $i=2,3,\ldots ,n$.

Theorem 3.22

The graph $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{s}_{3}$ satisfies the property (-SR).

Proof

The characteristic polynomial of the adjacency matrix of $\Gamma \in {{\mathcal {N}}}{{\mathcal {C}}}^{s}_{3}$ is

$$\begin{aligned} {\small \det \Big (xI-A(\Gamma )\Big )=\det \left( \begin{array}{cc|cccc} xI_{k_{1}}-H&{}{{\textbf {0}}}_{k_{1},k_{2}}&{}-E(H)&{}{{\textbf {0}}}_{k_{1},n}&{}{{\textbf {0}}}_{k_{1},m}&{}{{\textbf {0}}}_{k_{1},n}\\ {{\textbf {0}}}_{k_{2},k_{1}}&{}xI_{k_{2}}-F&{}{{\textbf {0}}}_{k_{2},m}&{}-J_{k_{2},n}&{}{{\textbf {0}}}_{k_{2},m}&{}{{\textbf {0}}}_{k_{1},n}\\ \hline -E(H)^{T}&{}{{\textbf {0}}}_{n,k_{2}}&{}xI_{m}-A_{s}&{}-B&{}-I_{m}&{}{{\textbf {0}}}_{m,n}\\ {{\textbf {0}}}&{}-J_{n,k_{2}}&{}-B^{T}&{}xI_{n}-C_{s}&{}{{\textbf {0}}}_{n,m}&{}-I_{n}\\ {{\textbf {0}}}_{m,k_{1}}&{}{{\textbf {0}}}_{m,k_{2}}&{}-I_{m}&{}{{\textbf {0}}_{m,n}}&{}xI_{m}&{}{{\textbf {0}}}_{m,n}\\ {{\textbf {0}}}_{n,k_{1}}&{}{{\textbf {0}}}_{n,k_{2}}&{}{{\textbf {0}}_{n,m}}&{}-I_{n}&{}{{\textbf {0}}}_{m,n}&{}xI_{n}\\ \end{array}\right) ,} \end{aligned}$$

where H and F are as defined in Remark 3.17.

Applying Lemma 3.3, we get

$$\begin{aligned} \det \Big (xI-A(\Gamma )\Big )&=\det (xI_{k_{1}}-H)\det (xI_{k_{2}}-F)\nonumber \\&\quad \times \det \left( \begin{array}{cccc} xI_{m}-(A_{s}+aJ_{m})&{}-B&{}-I_{m}&{}{{\textbf {0}}}_{m,n}\\ -B^{T}&{}xI_{n}-(C_{s}+bJ_{n})&{}{{\textbf {0}}}_{n,m}&{}-I_{n}\\ -I_{m}&{}{{\textbf {0}}}_{m,n}&{}xI_{m}&{}{{\textbf {0}}}_{m,n}\\ { {\textbf {0}}}_{n,m}&{}-I_{n}&{}{ {\textbf {0}}}_{m,n}&{}xI_{n} \end{array}\right) , \end{aligned}$$

(3.6)

where $a=\displaystyle \sum _{i=1}^{k}E^{T}_{i}\Big (xI_{4p_{i}}-A(H_{i})\Big )^{-1}E_{i}$ and $b=\displaystyle \sum _{j=1}^{\ell }{} {\textbf {1}}^{T}_{2m_{j}}\Big (xI_{2m_{j}}-A(F_{j})\Big )^{-1}{} {\textbf {1}}_{2m_{j}}$. By Lemmas 3.5 and 3.7, we obtain

$$\begin{aligned} a=\dfrac{k_{1}x(x^{2}-1)}{2(x^{4}-6x+1)}\quad \text{ and }\quad b=\dfrac{k_{2}x}{x^{2}-2x-1}. \end{aligned}$$

Applying Lemma 3.3 to Eq. (3.6), we see that

$$\begin{aligned}&\det \Big (xI-A(\Gamma )\Big )\\&=x^{n+m}\det (xI_{k_{1}}-H)\det (xI_{k_{2}}-F)\\&\quad \times \det \left( \begin{array}{cccc} \left( x-\dfrac{1}{x}\right) I_{m}-(A_{s}+aJ_{m})&{}-B\\ -B^{T}&{}\left( x-\dfrac{1}{x}\right) I_{n}-(C_{s}+bJ_{n}) \end{array} \right) , \end{aligned}$$

By Remark 3.21,

$$\begin{aligned}&\det \Big (xI-A(\Gamma )\Big )\nonumber \\&\quad =x^{n+m}\det (xI_{k_{1}}-H)\det (xI_{k_{2}}-F)\nonumber \\&\qquad \times \det \left( \begin{array}{cccc} \left( x-\dfrac{1}{x}\right) I_{m}-A^{\prime }_{s}&{}-B^{\prime }\\ {-B^{\prime }}^{T}&{}\left( x-\dfrac{1}{x}\right) I_{n}-C^{\prime }_{s} \end{array}\right) \nonumber \\&\quad =x^{n+m}\det (xI_{k_{1}}-H)\det (xI_{k_{2}}-F)\det \left( \left( x-\dfrac{1}{x}\right) I_{n}-C^{\prime }_{s}\right) \nonumber \\&\qquad \times \det \left( \left( x-\dfrac{1}{x}\right) I_{m}-A^{\prime }_{s}-B^{\prime }\left( \left( x-\dfrac{1}{x}\right) I_{n}-C^{\prime }_{s}\right) ^{-1}{B^{\prime }}^{T} \right) . \end{aligned}$$

(3.7)

Case I: $s=1$. From Eq. (3.7) and by Remark 3.21, we obtain

$$\begin{aligned} P(\Gamma ;x)&=\dfrac{(x^2-1)^{m-n}}{2(x^{2}-2x-1)(x^4-6x^2+1)}\prod _{i=1}^{k}P(H_{i};x)\\&\quad \times \prod _{j=1}^{\ell }P(F_{j};x)\prod _{i=2}^{n}(x^4-(r+\lambda _{i}+2)x^2+1)R_{1}(x), \end{aligned}$$

where $R_{1}(x)=2x^{10}-4x^9-(k_{1}m+2k_{2}n+4r+18)x^8+(2k_{1}m+8r+32)x^7+(k_{1}k_{2}mn+3k_{1}m+14k_{2}n+28r+44)x^6+(-4k_{1}m-48r-56)x^5-(k_{1}k_{2}mn+3k_{1}m+14k_{2}n+28r+44)x^4+(2k_{1}m+8r+32)x^3+(k_{1}m+2k_{2}n+4r+18)x^2-4x-2$.

Now, the result follows for $s=1$, by Lemma 3.1.

Case II: $s=2$. From Eq. (3.7) and by Remark 3.21, we obtain

$$\begin{aligned} P(\Gamma ;x)&=\dfrac{(x^2-1)^{m-n}}{2(x^{2}-2x-1)(x^4-6x^2+1)}\prod _{i=1}^{k}P(H_{i};x)\,\prod _{j=1}^{\ell }P(F_{j};x)\\&\quad \times \prod _{i=2}^{n}\,\Big (x^4-(\lambda _{i}+r-2)x^3-(r+\lambda _{i}+2)x^2\\&\quad -(-\lambda _{i}-r+2)x+1\Big )\,R_{2}(x), \end{aligned}$$

where $R_{2}(x)=2x^{10}-4rx^9+(-k_{1}m-2k_{2}n+4r-26)x^8+(4k_{2}nr+2k_ {1}m-4k_{2}n+40r)x^7+(k_{1}k_{2}mn+3k_{1}m+14k_{2}n-28r+100)x^6+ (-24k_{2}nr-4k_{1}m+24k_{2}n-104r)x^5+(-k_{1}k_{2}mn-3k_{1}m-14k_ {2}n+28r-100)x^4+(4k_{2}nr+2k_{1}m-4k_{2}n+40r)x^3+(k_{1}m+2k_{2}n-4r+26)x^2-4rx-2$.

Now, the result follows for $s=2$, by Lemma 3.1.

Case III: $s=3$. From Eq. (3.7) and by Remark 3.21, we obtain

$$\begin{aligned} P(\Gamma ;x)&=\dfrac{(x^2-1)^{m-n}}{2(x^{2}-2x-1)(x^4-6x^2+1)}\prod _{i=1}^{k}P(H_{i};x)\\&\quad \times \prod _{j=1}^{\ell }P(F_{j};x)\prod _{i=2}^{n}(x^4-\lambda _{i}x^3-(r+\lambda _{i}+2)x^2+\lambda _{i}x+1)R_{3}(x), \end{aligned}$$

where $R_{3}(x)=2x^{10}+(-2r-4)x^9+(-k_{1}m-2k_{2}n-18)x^8+(k_{1}mr+2k_{1}m+24r+32)x^7+(k_{1}k_{2}mn-2k_{1}mr+3k_{1}m+14k_{2}n+44)x^6+(-2k_{1}mr-4k_{1}m-76r-56)x^5+(-k_{1}k_{2}mn+2k_{1}mr-3k_{1}m-14k_{2}n-44)x^4+(k_{1}mr+2k_{1}m+24r+32)x^3+(k_{1}m+2k_{2}n+18)x^2+(-2r-4)x-2$.

Now, the result follows for $s=3$, by Lemma 3.1.

Case IV: $s=4$. From Eq. (3.7) and by Remark 3.21, we obtain

$$\begin{aligned} P(\Gamma ;x)&=\dfrac{(x^2-1)^{m-n}}{2(x^{2}-2x-1)(x^4-6x^2+1)}\prod _{i=1}^{k}P(H_{i};x)\prod _{j=1}^{\ell }P(F_{j};x)\\&\quad \times \prod _{i=2}^{n}\left( x^4-(r+2\lambda _{i}-2)x^3+\left( \lambda ^2_{i}+r\lambda _{i}-3\lambda _{i}-r-2\right) x^2\right. \\&\quad \left. +(r+2\lambda _{i}-2)x+1\right) R_{4}(x), \end{aligned}$$

where $R_{4}(x)=2x^{10}-6rx^9+(-k_{1}m-2k_{2}n+4r^2+4r-26)x^8+(k_{1}mr +4k_{2}nr+2k_{1}m-4k_{2}n-8r^2+64r)x^7+(k_{1}k_{2}mn-2k_{1}mr+3k_ {1}m+14k_{2}n-28r^2-28r+100)x^6+(-2k_{1}mr-24k_{2}nr-4k_{1}m+24k_ {2}n+48r^2-180r)x^5+(-k_{1}k_{2}mn+2k_{1}mr-3k_{1}m-14k_{2}n+28r^2+28r-100)x^4 +(k_{1}mr+4k_{2}nr+2k_{1}m-4k_{2}n-8r^2+64r)x^3+(k_{1}m+2k_{2}n-4r^2-4r+26)x^2-6rx-2$. Now, the result follows for $s=4$, by Lemma 3.1. $\square$

4 Conclusion

The paper is aimed to construct classes of noncorona graphs which satisfy property (-SR). Employing the corona product of an even cycle and an isolated vertex, we have given several classes of noncorona graphs which satisfy property (-SR). Thus, we have extended the few known class of noncorona graphs which satisfy property (-SR). Also, we have proved that there is no singular unicyclic graph whose nonzero eigenvalues satisfy reciprocal eigenvalue property. It would be interesting to see whether the result is true for any singular graph.