The Generalized Characteristic Polynomial of the K_m,n-Complement of a Bipartite Graph

Abstract

The generalized matrix of a graph G is defined as $M\left(G\right)=A\left(G\right)-tD\left(G\right)$ ($t\in \mathbb{R}$, and $A\left(G\right)$ and $D\left(G\right)$, respectively, denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of $M\left(G\right)$. Let $K_{m,n}$ be the complete bipartite graph. Then, the $K_{m,n}$-complement of a subgraph G in $K_{m,n}$ is defined as the graph obtained by removing all edges of an isomorphic copy of G from $K_{m,n}$. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the $K_{m,n}$-complement of a bipartite subgraph G is provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the $K_{m,n}$-complements of G is given.

1. Introduction

In the present paper, we only consider undirected, simple, and connected graphs unless otherwise stated. Let $G=\left(V\right(G),E(G\left)\right)$ (or shortly $(V,E)$) be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. The adjacency matrix of G is defined as $A\left(G\right)=\left(a_{ij}\right)$, where $a_{ij}$ equals the number of edges connecting vertices $v_i$ and $v_j$ when $i\ne j$ and 0 when $i=j$. The rank of a graph G, denoted by $r\left(G\right)$, is defined to be the rank of its adjacency matrix $A\left(G\right)$. The degree matrix $D\left(G\right)$ is defined as the diagonal matrix $\mathrm{diag}(d_1,d_2,\dots ,d_n)$, where $n=\left|V\right|$, and $d_i$ equals the number of edges incident to vertex $v_i$. In the literature [1], Cvetković et al. introduced a bivariate polynomial, $\varphi (G;\lambda ,t)=det(\lambda I-(A\left(G\right)-tD\left(G\right)\left)\right)$ (abbreviated as $\varphi \left(G\right)$). Wang et al. [2] referred to it as the generalized characteristic polynomial of G. It is natural to define $A\left(G\right)-tD\left(G\right)$ in the variable t as the generalized matrix of a graph G, denoted by $M\left(G\right)={\left(m_{ij}\left(G\right)\right)}_{\left|V\right|\times \left|V\right|}$. To be specific, it is easy to see that

$$m_{ij}\left(G\right)=\left\{\begin{array}{cc}-t{d}_G\left(v_i\right)\hfill & \mathrm{if}i=j,\hfill \\ 1\hfill & \mathrm{if}i\ne j\mathrm{and}{v}_i\sim v_j,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Consequently, the generalized characteristic polynomial of graph G is exactly the characteristic polynomial of the generalized matrix $M\left(G\right)$. That is, the polynomial $\varphi \left(G\right)=\varphi (G;\lambda ,t)=det(\lambda I-M(G\left)\right)=det(\lambda I-(A\left(G\right)-tD\left(G\right)\left)\right)$ is referred to as the generalized characteristic polynomial of G. Note that $A\left(G\right)-tD\left(G\right)$ with $t\in \mathbb{R}$ encodes several well-known graph matrices, such as the adjacency matrix, the Laplacian matrix, and the normalized Laplacian matrix. It is evident that the generalized characteristic polynomial of a graph generalizes several well-known polynomial invariants of graphs, and the following are some examples:

• The characteristic polynomial of the adjacency matrix of a graph G is given by $\varphi (G;\lambda ,0)=det(\lambda I-A(G\left)\right)$;
• The characteristic polynomial of the Laplacian matrix $D\left(G\right)-A\left(G\right)$ of G is
  ${(-1)}^{\left|V\right|}\varphi (G;-\lambda ,1)=det(-\lambda I-A\left(G\right)+D\left(G\right))$;
• The characteristic polynomial of the unsigned Laplacian matrix $D\left(G\right)+A\left(G\right)$ of graph G is $\varphi (G;\lambda ,-1)=det(\lambda I-A(G)-D(G\left)\right)$;
• The characteristic polynomial of the normalized Laplacian matrix $I-D_G^{-{\displaystyle \frac{1}{2}}}{A}_G{D}_G^{-{\displaystyle \frac{1}{2}}}$ is ${(-1)}^{\left|V\right|}\varphi (G;0,-\lambda +1)=det(-A\left(G\right)-(\lambda -1)D\left(G\right))$.

Given an undirected graph $G=(V,E)$, if E is considered as a set of symmetric directed edges, meaning that if $e\in E$, then $\overline{e}\in E$, where $\overline{e}$ is the reverse edge of e, then G can also be viewed as a directed graph. For $e\in E$, let $h\left(e\right)$ denote the head of the directed edge e and $t\left(e\right)$ the tail of e. A closed walk in G is defined as a sequence of edges $C=(e_1,\dots ,e_k)$ such that $h\left(e_i\right)=t\left(e_{i+1}\right)$ for $i\in \mathbb{Z}/k\mathbb{Z}$. Here, $k=\left|C\right|$ is the length of C and $cbc\left(C\right)=\#\{i\in \{1,\dots ,k\}\mid e_{i+1}=\overline{e_i}\}$ is called the cyclic bump count of C. The notation $\left[C\right]$ is referred to as the equivalence class of the closed walk C under edge permutation, meaning that $(e_1,\dots ,e_k)\sim (e_2,\dots ,e_k,e_1)$. If none of the representatives of $\left[C\right]$ can be expressed as $C^k$ (for $k\ge 2$), then the cycle C is said to be irreducible. The set of all irreducible cycles is denoted by $\mathcal{C}$. The Bartholdi zeta function of a graph G is defined as (see [3] for details)

$$Z_G(\lambda ,t)=\prod _{\left[C\right]\in \mathcal{C}}{\displaystyle \frac{1}{1-{\lambda }^{cbc\left(C\right)}{t}^{\left|C\right|}}}.$$

The function $Z_G\left(t\right)=Z_G(0,t)$ is referred to as the (Ihara–Selberg) zeta function [4,5], which was introduced by Ihara to study the zeta function of a regular graph and its reciprocal, and the reciprocal of the zeta function of a regular graph was generalized to the reciprocal of the Bartholdi zeta function for a general graph G as below:

$$Z_G{(\lambda ,t)}^{-1}={\left(1-{(1-\lambda )}^2{t}^2\right)}^{\left|E\right|-\left|V\right|}det\left(I-t{A}_G+(1-\lambda )(D_G-(1-\lambda )I)t^2\right).$$

In particular, the reciprocal of the zeta function for a general graph G is given by

$$Z_G{\left(t\right)}^{-1}={(1-t^2)}^{\left|E\right|-\left|V\right|}det\left(I-t{A}_G+t^2(D_G-I)\right).$$

The zeta function encodes significant structural information about the graph, such as the number of vertices, edges, and loops. Moreover, the number of spanning trees (the complexity of the graph) $\tau \left(G\right)$ satisfies the following equation [6]:

$${\displaystyle \frac{\partial f_G\left(t\right)}{\partial t}}{|}_{t=1}=2\left(\right|E|-\left|V\right|)\tau \left(G\right),\mathrm{where}{f}_G\left(t\right)=det\left(I-t{A}_G+t^2(D_G-I)\right).$$

For a comprehensive treatment of many aspects of the zeta function, refer to [7].

Zeta functions of certain classes of graphs have received considerable attention, such as the line graph of semi-regular bipartite graphs [8], the middle graph of semi-regular bipartite graphs [9], the cone graph of regular graphs [10] or semi-regular graphs [11], and various special join graphs of regular graphs [12,13]. It is not difficult to prove that $\varphi (G;\lambda ,t)$ determines the reciprocal of the zeta function, and vice versa [14]. Let $G=(V,E)$ be a subgraph of the complete bipartite graph $K_{m,n}$. The $K_{m,n}$-complement of G is defined as the graph obtained from $K_{m,n}$ by deleting all edges of G in $K_{m,n}$, i.e., $K_{m,n}-E\left(G\right)$. In this paper, we shall show a computational method for deriving the formula for the generalized characteristic polynomial of the $K_{m,n}$-complement of any bipartite graph G, and further give an explicit formula for the generalized characteristic polynomials of the $K_{m,n}$-complement of a bipartite graph with rank less than or equal to 4.

2. Notations and Terminology

Let $G=(V,E)$ be a graph with the vertex set $V=\{v_1,v_2,\dots ,v_n\}$ and the edge set $E=\{e_1,\dots ,e_m\}$. For two vertices $v_i,v_j\in V$, if $v_i$ and $v_j$ are adjacent, we denote this as $v_i\sim v_j$. The neighborhood of a vertex $v_i$ in G is defined as $N_G\left(v_i\right)=\{v_j\in V\mid v_i\sim v_j\}$, and the degree of vertex $v_i$ in G is denoted by $d_i=d_G\left(v_i\right)=\left|N_G\left(v_i\right)\right|$. The complement of the graph $G=(V,E)$ is denoted as $G^c=(V,E^c)$, where $E^c=\{v_i{v}_j\mid v_i,v_j\in V,v_i{v}_j\notin E\}$. If $G=(V,E)$ and $G^{\prime }=(V^{\prime },E^{\prime })$ with $V^{\prime }\subseteq V$ and $E^{\prime }=\{(u,v)\mid u,v\in V^{\prime },(u,v)\in E\}$, then $G^{\prime }$ is referred to as an induced subgraph of G.

A graph $G=(V,E)$ is a bipartite graph if and only if there exists a bipartition of V into $(V_1,V_2)$ (namely $V=V_1\cup V_2,V_1\cap V_2=\mathsf{Ø}$) such that no two vertices within $V_1$ or $V_2$ are adjacent. If the sizes of the bipartition sets are equal, i.e., $|V_1|=|V_2|$, then G is said to be a balanced bipartite. If $G=(V,E)$ is a bipartite graph with a bipartition $(V_1,V_2)$, the bipartite complement of G, denoted as $G^{bc}$, has the vertex set $V\left(G^{bc}\right)=V\left(G\right)$ and edge set $E\left(G^{bc}\right)=\{xy\mid x\in V_1,y\in V_2,xy\notin E\left(G\right)\}$. For a bipartite graph G, its adjacency matrix is given by

$$A\left(G\right)=\left[\begin{array}{cc}0& B\left(G\right)\\ B^T\left(G\right)& 0\end{array}\right],$$

where $B\left(G\right)={\left(b_{ij}\right)}_{m\times n}$ is the bipartite adjacency matrix of G that defines the vertex adjacency relationship between the bipartite sets $V_1$ and $V_2$. Specifically,

$$b_{ij}=\left\{\begin{array}{cc}1\hfill & v_i\sim v_j,v_i\in V_1,v_j\in V_2\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Lemma 1

([15]). Let G be a balanced bipartite graph. If G has a unique perfect matching, then the bipartite adjacency matrix $B\left(G\right)$ has determinant 1 or $-1$.

Lemma 2

([16]). Let $A={\left(a_{ij}\right)}_{n\times n}$ be an $n\times n$ matrix, and let D be a diagonal matrix with diagonal entries $d_1,d_2,\dots ,d_n$, i.e., $D=\mathrm{diag}(d_1,d_2,\dots ,d_n)$. Then,

$$det(A+D)=\sum _{\theta \subseteq \left[n\right]}det\left(A_{\theta }\right)det\left(D_{\overline{\theta }}\right),$$

(1)

where θ is a subset of $\left[n\right]=\{1,2,\dots ,n\}$ and $\overline{\theta }$ is the complement of θ in $\left[n\right]$, namely $\overline{\theta }=\{k\mid k\in \left[n\right],k\notin \theta \}$; $A_{\theta }$ is the submatrix formed by the rows and columns of A indexed by θ. By convention, $det\left(A_{\mathsf{Ø}}\right)=1$.

The following lemma immediately follows from Lemma 2.

Lemma 3

([16]). If D is an invertible matrix, then the determinant of the matrix $A+D$ can be expressed as

$$det(A+D)=det\left(D\right)\sum _{\theta \subseteq \left[n\right]}{\displaystyle \frac{det\left(A_{\theta }\right)}{det\left(D_{\theta }\right)}}.$$

(2)

Lemma 4

([17]). Let A be an $n\times n$ matrix. If there exists a $p\times q$ zero submatrix in A such that $p+q\ge n+1$, then $det\left(A\right)=0.$

3. The Generalized Characteristic Polynomial

For the sake of simplicity, the complete graph, cycle, and path on n vertices are denoted by $K_n$, $C_n$, and $P_n$, respectively. Notationally, for $m,n\in \mathbb{Z}$, $\left[m\right]=\{1,2,\dots ,m\}$, and $[m+1,m+n]=\{m+1,m+2,\dots ,m+n\}$; $I_n$ and $J_{m\times n}$ (or $J_n$), respectively, denote the $n\times n$ identity matrix and the $m\times n$ (or $n\times n$) matrix of all ones. For the rest of this paper, we will use $K_{m,n}$ to symbolize the complete bipartite graph with bipartite partition $(X,Y)$, where $X=\{v_1,v_2,\dots ,v_m\}$ and $Y=\{v_{m+1},v_{m+2},\dots ,v_{m+n}\}$, and ${\mathcal{B}}_{m,n}$ to symbolize the set of all bipartite graphs with bipartite partition $(X,Y)$ such that $\left|X\right|=m$ and $\left|Y\right|=n$. Note that a graph $G\in {\mathcal{B}}_{m,n}$ if and only if its bipartite complement $G^{bc}\in {\mathcal{B}}_{m,n}$. Let G be a subgraph of the complete bipartite graph $K_{m,n}$. The $K_{m,n}$-complement of a subgraph G in $K_{m,n}$ is defined as the graph obtained from $K_{m,n}$ by deleting all edges of G in $K_{m,n}$, denoted by $K_{m,n}-G$.

Theorem 1.

Let $G=(V,E)$ be a subgraph of $K_{m,n}$ with bipartite partition $(X,Y)$ mentioned previously, and G has a bipartite partition $(V_1,V_2)$ such that $V_1\subseteq X$ ($|V_1|=s$), $V_2\subseteq Y$ ($|V_2|=t$). Then, we have

$$\begin{array}{cc}\hfill & \varphi (K_{m,n}-G)\hfill \\ \hfill & ={(nt+\lambda )}^{m-s}{(mt+\lambda )}^{n-t}{\displaystyle \prod _{v\in V_1}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V_2}}((m-d_G\left(v\right))t+\lambda )\hfill \\ \hfill & ·\left[1+{\displaystyle \sum _{Q^{bc}\in \mathcal{G}\left(H^{bc}\right)}}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q^{bc}\left)\right|}{2}}{\left(det(J_k-B\left(Q^{bc}\right))\right)}^2}{{\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}\right]\hfill \\ \hfill & ={(nt+\lambda )}^{m-s}{(mt+\lambda )}^{n-t}{\displaystyle \prod _{v\in V_1}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V_2}}((m-d_G\left(v\right))t+\lambda )\hfill \\ \hfill & ·\left[1+{\displaystyle \sum _{Q\in \mathcal{G}(K_{m,n}-G)}}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q\left)\right|}{2}}{\left(det\left(B\right(Q\left)\right)\right)}^2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}\right],\hfill \end{array}$$

(3)

where $\mathcal{G}\left(H^{bc}\right)$ is the set of all induced balanced bipartite subgraphs $Q^{bc}$ in H such that the bipartite complement Q of $Q^{bc}$ has a nonsingular biadjacency matrix $B\left(Q\right)$; $\mathcal{G}\left(H\right)$ is the set of all nonempty induced balanced bipartite subgraphs Q in H (i.e., $K_{m,n}-G$) such that $B\left(Q\right)$ is nonsingular.

Proof. 

The proof is based on the use of fundamental linear algebra and an expansion of the determinant on the sum of two square matrices (see Lemma 2); the diagonal one of two matrices is highly related to the degree matrix of the graph $K_{m,n}-G$, and the other is a variant of the adjacency matrix of the graph $K_{m,n}-G$. One trick of the proof lies in the fact that half of the principal submatrices of the non-diagonal variant have determinant zero, which leads to a further simplification of the computation.

Let $H=K_{m,n}-G\in {\mathcal{B}}_{m,n}$ (here, ${\mathcal{B}}_{m,n}$ denotes the set of all bipartite graphs with bipartite partition $(X,Y)$ such that $\left|X\right|=m$ and $\left|Y\right|=n$). Note that H has the same bipartite partition as $K_{m,n}$, that is, $(X,Y)=\{v_1,v_2,\dots ,v_m\}\cup \{v_{m+1},\dots ,v_{m+n}\}$, where $V_1\subseteq X$ and $V_2\subseteq Y$. Obviously, $H^{bc}\cong G\cup (m+n-|V\left|\right)K_1\in {\mathcal{B}}_{m,n}$. The generalized matrix $M\left(H\right)=A\left(H\right)-tD\left(H\right)={\left(m_{ij}\left(H\right)\right)}_{(m+n)\times (m+n)}$ is given by

$$m_{ij}\left(H\right)=\left\{\begin{array}{cc}t(d_{H^{bc}}\left(v_i\right)-n)\hfill & \mathrm{if}i=j\in \left[m\right],\hfill \\ t(d_{H^{bc}}\left(v_i\right)-m)\hfill & \mathrm{if}i=j\in [m+1,m+n],\hfill \\ 1\hfill & \mathrm{if}i\ne j\mathrm{and}{v}_i{v}_j\in E\left(H\right),\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In other words,

$$M\left(H\right)=\left[\begin{array}{cc}-t{D}_1\left(H\right)& J_{m\times n}-B\left(H^{bc}\right)\\ J_{n\times m}-B^T\left(H^{bc}\right)& -t{D}_2\left(H\right)\end{array}\right]$$

where $D_1\left(H\right)=\mathrm{diag}(n-d_{H^{bc}}\left(v_1\right),n-d_{H^{bc}}\left(v_2\right),\dots ,n-d_{H^{bc}}\left(v_m\right))$ and $D_2\left(H\right)=\mathrm{diag}(m-d_{H^{bc}}\left(v_{m+1}\right),m-d_{H^{bc}}\left(v_{m+2}\right),\dots ,m-d_{H^{bc}}\left(v_{m+n}\right))$.

Let ${\mathbf{1}}_{V_1}$ be the $(m+n)\times 1$ column vector such that the i-th element is 1 for $1\le i\le m$ and 0 for $m+1\le i\le m+n$. Similarly, let ${\mathbf{1}}_{V_2}$ be the $(m+n)\times 1$ vector where the i-th element is 0 for $1\le i\le m$ and 1 for $m+1\le i\le m+n$.

$$\begin{array}{ccc}\lambda I_{m+n}-M\left(H\right)\hfill & =\hfill & \left[\begin{array}{cc}t{D}_1\left(H\right)+\lambda I_m& B\left(H^{bc}\right)-J_{m\times n}\\ B^T\left(H^{bc}\right)-J_{n\times m}& t{D}_2\left(H\right)+\lambda I_n\end{array}\right]\hfill \\ \hfill & =\hfill & tD\left(H\right)+\lambda I_{m+n}+A\left(H^{bc}\right)-{\mathbf{1}}_{V_1}{\mathbf{1}}_{V_2}^T-{\mathbf{1}}_{V_2}{\mathbf{1}}_{V_1}^T\hfill \\ \hfill & =\hfill & D^{\prime }+A^{\prime }\hfill \end{array}$$

Let $D^{\prime }=tD\left(H\right)+\lambda I_{m+n}$ and $A^{\prime }=A\left(H^{bc}\right)-{\mathbf{1}}_{V_1}{\mathbf{1}}_{V_2}^T-{\mathbf{1}}_{V_2}{\mathbf{1}}_{V_1}^T$. Then, we have

$$det\left(D^{\prime }\right)=\prod _{v\in X}((n-d_{H^{bc}}\left(v\right))t+\lambda )\prod _{v\in Y}((m-d_{H^{bc}}\left(v\right))t+\lambda )\ne 0.$$

By Lemma 2, the following equality holds:

$$det(\lambda I_{m+n}-M\left(H\right))=det(D^{\prime }+A^{\prime })=det\left(D^{\prime }\right)\sum _{\theta \subseteq [m+n]}{\displaystyle \frac{det\left(A_{\theta }^{\prime }\right)}{det\left(D_{\theta }^{\prime }\right)}}.$$

(4)

Now, we claim two facts:

Fact 1: The first-order principal submatrix of $A^{\prime }$ is zero. The third-order principal submatrices of $A^{\prime }$ are in the form

$$\left[\begin{array}{ccc}0& 0& \ast \\ 0& 0& \ast \\ \ast & \ast & 0\end{array}\right]or\left[\begin{array}{ccc}0& \ast & \ast \\ \ast & 0& 0\\ \ast & 0& 0\end{array}\right],$$

and both of their determinants are zero. The fifth-order principal submatrices of $A^{\prime }$ are in one of the following forms:

$$\left[\begin{array}{ccccc}0& \ast & \ast & \ast & \ast \\ \ast & 0& 0& 0& 0\\ \ast & 0& 0& 0& 0\\ \ast & 0& 0& 0& 0\\ \ast & 0& 0& 0& 0\end{array}\right],\left[\begin{array}{ccccc}0& 0& \ast & \ast & \ast \\ 0& 0& \ast & \ast & \ast \\ \ast & \ast & 0& 0& 0\\ \ast & \ast & 0& 0& 0\\ \ast & \ast & 0& 0& 0\end{array}\right],\left[\begin{array}{ccccc}0& 0& 0& \ast & \ast \\ 0& 0& 0& \ast & \ast \\ 0& 0& 0& \ast & \ast \\ \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & 0& 0\end{array}\right],\left[\begin{array}{ccccc}0& 0& 0& 0& \ast \\ 0& 0& 0& 0& \ast \\ 0& 0& 0& 0& \ast \\ 0& 0& 0& 0& \ast \\ \ast & \ast & \ast & \ast & 0\end{array}\right],$$

and each of their determinants equals zero. Analogously, the odd-order principal submatrices of $A^{\prime }$ are in the form

$$A_{\theta }^{\prime }=\left[\begin{array}{cc}{0}_{p\times p}& A_1\\ A_1^T& 0_{q\times q}\end{array}\right](p+q=|\theta \left|\right).$$

Note that $2p\ge n+1$ or $2q\ge n+1$. According to Lemma 4, we conclude that $det\left(A_{\theta }^{\prime }\right)=0$, indicating that all odd-order principal submatrices of $A^{\prime }$ are singular.

Fact 2: By definition, $A^{\prime }=A\left(H^{bc}\right)-{\mathbf{1}}_{V_1}{\mathbf{1}}_{V_2}^T-{\mathbf{1}}_{V_2}{\mathbf{1}}_{V_1}^T$, or equivalently

$$A^{\prime }=\left[\begin{array}{cc}0& B\left(H^{bc}\right)-J_{m\times n}\\ B^T\left(H^{bc}\right)-J_{n\times m}& 0\end{array}\right].$$

Suppose $\theta$ is a subset of $[m+n]$ such that ${\displaystyle \frac{\left|\theta \right|}{2}}=|\theta \cap \left[m\right]|=|\theta \cap [m+1,m+n]|$. Let $A_{\theta }^{\prime }$ be an even-order principal submatrix of $A^{\prime }$. We denote by $A_{\theta \cap \left[m\right],\theta \cap [m+1,m+n]}^{\prime }$ the submatrix of $A^{\prime }$ formed by the rows indexed by $\theta \cap \left[m\right]$ and the columns indexed by $\theta \cap [m+1,m+n]$. Furthermore,

$$det\left(A_{\theta }^{\prime }\right)={(-1)}^{{\displaystyle \frac{\left|\theta \right|}{2}}\times {\displaystyle \frac{\left|\theta \right|}{2}}}{\left(det\left(A_{\theta \cap \left[m\right],\theta \cap [m+1,m+n]}^{\prime }\right)\right)}^2.$$

This reduces to

$$\begin{array}{cc}\hfill det\left(A_{\theta }^{\prime }\right)& ={(-1)}^{{\displaystyle \frac{\left|\theta \right|}{2}}\times {\displaystyle \frac{\left|\theta \right|}{2}}}{\left(det(B_{\theta }-J_{{\displaystyle \frac{\left|\theta \right|}{2}}})\right)}^2\hfill \\ \hfill & ={(-1)}^{{\left({\displaystyle \frac{\left|\theta \right|}{2}}\right)}^2}{\left(det(J_{{\displaystyle \frac{\left|\theta \right|}{2}}}-B_{\theta }\mathtt{)}\right)}^2\hfill \\ \hfill & ={(-1)}^{{\left({\displaystyle \frac{\left|\theta \right|}{2}}\right)}^2}{\left(det\left({\overline{B}}_{\theta }\right)\right)}^2\hfill \\ \hfill & ={(-1)}^{\left|\theta \right|}{\left(det\left({\overline{B}}_{\theta }\right)\right)}^2,\hfill \end{array}$$

where $B_{\theta }$ is the ${\displaystyle \frac{\left|\theta \right|}{2}}\times {\displaystyle \frac{\left|\theta \right|}{2}}$ submatrix obtained from $B\left(H^{bc}\right)$ by deleting the rows that are not in $\theta \cap \left[m\right]$ and the columns that are not in $\theta \cap [m+1,m+n]$. Similarly, ${\overline{B}}_{\theta }$ is the matrix resulting from $B\left(H\right)$ by the same deletion of rows and columns, and it is easy to see $B_{\theta }+{\overline{B}}_{\theta }=J_{{\displaystyle \frac{\left|\theta \right|}{2}}}$.

Set $k={\displaystyle \frac{\left|\theta \right|}{2}}$, and let Q be a balanced bipartite induced subgraph of H with $2k$ vertices, and the bipartite complement $Q^{bc}$ corresponding to Q is also a balanced bipartite induced subgraph of $H^{bc}$. Obviously, $B\left(Q\right)+B\left(Q^{bc}\right)=J_k$. Let ${\mathcal{G}}_k\left(H^{bc}\right)$ denote the set of induced balanced bipartite subgraphs $Q^{bc}$ in H with $2k$ vertices such that the bipartite complement Q of $Q^{bc}$ has a nonsingular biadjacency matrix $B\left(Q\right)$; that is, the rank of $B\left(Q\right)$ is k ($k\le r\left(A^{\prime }\right)/2$). Moreover, we denote by $\mathcal{G}\left(H^{bc}\right)$ the set of all induced balanced bipartite subgraphs $Q^{bc}$ in H such that $B\left(Q\right)$ is nonsigular. We denote by $\mathcal{G}\left(H\right)$ the set of all nonempty induced balanced bipartite subgraphs Q in H (i.e., $K_{m,n}-G$) such that $B\left(Q\right)$ is nonsingular; that is to say, if Q is such a nonempty induced balanced bipartite subgraph in H on $2k$ vertices, then the $k\times k$ matrix $B\left(Q\right)$ satisfies the condition $r(J_k-B\left(Q^{bc}\right))=k$, and it is worth noting that the induced subgraph Q may not necessarily be nonsingular.

Observe that (i) $r\left(A^{\prime }\right)=2r(J_{m\times n}-B\left(H^{bc}\right))=2r(B\left(H^{bc}\right)-J_{m\times n})\le 2r\left(B\left(H^{bc}\right)\right)+2r\left(J_{m\times n}\right)=2r\left(B\left(G\right)\right)+2=r\left(A\left(G\right)\right)+2$, where $r\left(M\right)$ symbolizes the rank of the matrix M; (ii) $B\left(Q\right)=J_k-B\left(Q^{bc}\right)$. Then, by Lemma 2 and Lemma 3 we conclude that

$$\begin{array}{cc}\hfill & \varphi (K_{m,n}-G)=det(D^{\prime }+A^{\prime })\hfill \\ \hfill =& \prod _{v\in X}((n-d_{H^{bc}}\left(v\right))t+\lambda )\prod _{v\in Y}((m-d_{H^{bc}}\left(v\right))t+\lambda )\hfill \\ \hfill & ·\left[1+\sum _{k=1}^{r\left(A\right(G\left)\right)/2+1}\sum _{Q^{bc}\in {\mathcal{G}}_k\left(H^{bc}\right)}{\displaystyle \frac{{(-1)}^{k^2}{\left(det(J_k-B\left(Q^{bc}\right))\right)}^2}{{\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap X}}((n-d_{H^{bc}}\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap Y}}((m-d_{H^{bc}}\left(v\right))t+\lambda )}}\right]\hfill \\ \hfill =& {(nt+\lambda )}^{m-s}{(mt+\lambda )}^{n-t}\prod _{v\in V_1}((n-d_G\left(v\right))t+\lambda )\prod _{v\in V_2}((m-d_G\left(v\right))t+\lambda )\hfill \\ \hfill & \times \left[1+\sum _{Q^{bc}\in \mathcal{G}\left(H^{bc}\right)}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q^{bc}\left)\right|}{2}}{\left(det(J_k-B\left(Q^{bc}\right))\right)}^2}{{\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q^{bc}\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}\right]\hfill \\ \hfill =& {(nt+\lambda )}^{m-s}{(mt+\lambda )}^{n-t}\prod _{v\in V_1}((n-d_G\left(v\right))t+\lambda )\prod _{v\in V_2}((m-d_G\left(v\right))t+\lambda )\hfill \\ \hfill & ·\left[1+\sum _{Q\in \mathcal{G}(K_{m,n}-G)}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q\left)\right|}{2}}{\left(det\left(B\right(Q\left)\right)\right)}^2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}\right].\hfill \end{array}$$

This completes the proof. □

Remark 1.

Suppose $G=(V,E)$ is a spanning subgraph of $K_{m,n}$ with the same bipartite partition $(X,Y)$. In this case, $K_{m,n}-G=G^{bc}$ is exactly the bipartite complement of the graph G. Following the proof of Theorem 1, it is not difficult to verify that

$$\begin{array}{ccc}\varphi (G^{bc},\lambda ,0)\hfill & =\hfill & {\lambda }^{m+n}\left[1+{\displaystyle \sum _{Q\in \mathcal{G}(K_{m,n}-G)}}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q\left)\right|}{2}}{\left(det\left(B\right(Q\left)\right)\right)}^2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}\lambda {\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}\lambda }}\right],\hfill \\ \varphi (G,\lambda ,0)\hfill & =\hfill & {\lambda }^{m+n}\left[1+{\displaystyle \sum _{Q\in \mathcal{G}\left(G\right)}}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q\left)\right|}{2}}{\left(det\left(B\right(Q\left)\right)\right)}^2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}\lambda {\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}\lambda }}\right],\hfill \end{array}$$

where $\mathcal{G}\left(H\right)$ is the set of all nonempty induced balanced bipartite subgraphs in H. When comparing the two characteristic polynomials, it is readily apparent that the disparities in their summation terms are remarkably significant. Generally speaking, attempting to discover the correlation between the eigenvalues of G and those of $G^{bc}$ with the assistance of Theorem 1 can be extremely challenging. This challenge persists unless the structures of both G and $G^{bc}$ are relatively simple. For the characteristic polynomial of $L\left(G\right)$ (i.e., $\mu (G;x)=det\left(xI-L\left(G\right)\right)$), it is well known that [18]

$$\mu (\overline{G};x)={(-1)}^{\left|G\right|-1}{\displaystyle \frac{x}{\left|G\right|-x}}\mu (G;|G|-x)(note:|G| is the order of G).$$

Relying on Theorem 1, we cannot find a similar relationship between the Laplacian characteristic polynomial of G and that of $G^{bc}$. Consequently, the primary significance of Theorem 1 resides in its provision of concrete computational methodologies.

4. An Application

Our main result in this section gives an application of Theorem 1. A permutation matrix is a square matrix obtained from the same size identity matrix by a permutation of rows. Two n by n matrices A and B are said to be permutationally equivalent if there exist n by n permutation matrices $P,Q$ such that $PAQ=B$.

Theorem 2.

With the notations mentioned in Theorem 1, if G has its rank $r\left(A\right(G\left)\right)\le 4$, then

$$\begin{array}{cc}\varphi (K_{m,n}-G)=\hfill & {(nt+\lambda )}^{m-s}{(mt+\lambda )}^{n-t}{\displaystyle \prod _{v\in V_1}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V_2}}((m-d_G\left(v\right))t+\lambda )\hfill \\ & \times [1+{\displaystyle \sum _{Q\in {\Omega }_{{\mathcal{F}}_1}(K_{m,n}-G)}}{\displaystyle \frac{1}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}\hfill \\ & +{\displaystyle \sum _{Q\in {\Omega }_{{\mathcal{F}}_2}(K_{m,n}-G)}}{\displaystyle \frac{-1}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v_i\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v_i\right))t+\lambda )}}\hfill \\ & +{\displaystyle \sum _{Q\in {\Omega }_{\left\{C_6\right\}}(K_{m,n}-G)}}{\displaystyle \frac{-2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}],\hfill \end{array}$$

(5)

where ${\Omega }_Q\left(G\right)$ is the set of all induced bipartite subgraphs of G that are isomorphic to the graph Q, and ${\Omega }_{\mathcal{F}}\left(G\right)$ is the set of all induced bipartite subgraphs of G that are isomorphic to certain graph in the family $\mathcal{F}$. Here, ${\mathcal{F}}_1=\{2K_2,Q_7\},$ ${\mathcal{F}}_2=\{K_2,3K_2,K_2\cup P_4,P_6,Q_3,Q_6\},$ where $Q_3,Q_6$ and $Q_7$ are illustrated in Figure 1, Figure 2 and Figure 3.

Proof. 

The main idea of the proof depends on a classification that takes any positive integer $k\le 4$ and is shown to produce all balanced and induced bipartite subgraphs, each of which has a non-degenerate biadjancency matrix of order ${\displaystyle \frac{k}{2}}$. It is worth noting that the rank of any of its induced subgraphs will not exceed k if the rank of a bipartite graph G is k. Suppose k is small and fixed; the number of distinct $n\times n$ biadjacency matrices with rank at most ${\displaystyle \frac{k}{2}}$ is finite up to graph isomorphism.

From the proof of Theorem 1, we know that

$$det(\lambda I_{m+n}-M\left(H\right))=det(D^{\prime }+A^{\prime })=det\left(D^{\prime }\right)\sum _{\theta \subseteq [m+n]}{\displaystyle \frac{det\left(A_{\theta }^{\prime }\right)}{det\left(D_{\theta }^{\prime }\right)}}$$

(6)

where $H=K_{m,n}-G$, $A^{\prime }=A\left(H^{bc}\right)-{\mathbf{1}}_{V_1}{\mathbf{1}}_{V_2}^T-{\mathbf{1}}_{V_2}{\mathbf{1}}_{V_1}^T$, or

$$\begin{array}{c}{A}^{\prime }=\left[\begin{array}{cc}0& B\left(H^{bc}\right)-J_{m\times n}\\ B^T\left(H^{bc}\right)-J_{n\times m}& 0\end{array}\right].\hfill \end{array}$$

Hence, we only need to study the summation

$$\sum _{\theta \subseteq [m+n]}{\displaystyle \frac{det\left(A_{\theta }^{\prime }\right)}{det\left(D_{\theta }^{\prime }\right)}}$$

(7)

or equivalently

$$1+\sum _{Q\in \mathcal{G}(K_{m,n}-G)}{\displaystyle \frac{{(-1)}^{\frac{\left|V\right(Q\left)\right|}{2}}{\left(det\left(B\right(Q\left)\right)\right)}^2}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_G\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_G\left(v\right))t+\lambda )}}].$$

(8)

Observe that $r\left(A^{\prime }\right)=2r(B\left(H^{bc}\right)-J_{m\times n})\le 2r\left(B\left(H^{bc}\right)\right)+2r\left(J_{m\times n}\right)=2r\left(B\left(G\right)\right)+2=r\left(A\left(G\right)\right)+2\le 6$. The following cases need to be discussed:

Case 1: If $\theta =\mathsf{Ø}$, then

$${\displaystyle \frac{det\left(A_{\theta }^{\prime }\right)}{det\left(D_{\theta }^{\prime }\right)}}=1.$$

Case 2: If $\left|\theta \right|=2$, then

$$A_{\theta }^{\prime }=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$$

Consequently, $Q\cong K_2$ and $Q^{bc}\cong K_2^{bc}$ (the trivial graph on two vertices). The contribution of this case to the summation part of Equation (7) is given by

$${\displaystyle \sum _{Q\in {\Omega }_{K_2}(K_{m,n}-G)}}{\displaystyle \frac{-1}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_Q\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_Q\left(v\right))t+\lambda )}}.$$

Case 3: If $\left|\theta \right|=4$, then $|\theta \cap [m\left]\right|=|\theta \cap [m+1,m+n\left]\right|=2$. Let $\theta =\{i,j,k,l\}$ with $i<j<k<l$, where $i,j\in \left[m\right]$ and $k,l\in [m+1,m+n]$. In this case, the matrix $A^{\prime }$ is in the form

$${A^{\prime }}_{\theta }=\left[\begin{array}{cccc}0& 0& a_{ik}& a_{il}\\ 0& 0& a_{jk}& a_{jl}\\ a_{ik}& a_{jk}& 0& 0\\ a_{il}& a_{jl}& 0& 0\end{array}\right],$$

where $\left[\begin{array}{cc}{a}_{ik}& a_{il}\\ a_{jk}& a_{jl}\end{array}\right]\mathrm{is}\mathrm{permutationally}\mathrm{equivalent}\mathrm{to}\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\mathrm{or}\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ since $det\left(A_{\theta }^{\prime }\right)\ne 0$. This indicates that the induced subgraph Q in $K_{m,n}-G$ satisfies that $Q\cong 2K_2$ or $P_4$. The contribution of this case to the summation part of Equation (7) is given by

$$\sum _{Q\in {\Omega }_{\{2K_2,P_4\}}(K_{m,n}-G)}{\displaystyle \frac{1}{{\displaystyle \prod _{v\in V\left(Q\right)\cap V_1}}((n-d_Q\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap V_2}}((m-d_Q\left(v\right))t+\lambda )}}.$$

Case 4: If $\left|\theta \right|=6$, then $|\theta \cap [m\left]\right|=|\theta \cap [m+1,m+n\left]\right|=3$. Suppose $B_3\left(Q^{bc}\right)$ is the $3\times 3$ principal submatrix of $B\left(Q^{bc}\right)$ in $A^{\prime }$ such that

$$A_{\theta }^{\prime }=\left[\begin{array}{cc}0& B_3\left(Q^{bc}\right)-J_3\\ {(B_3\left(Q^{bc}\right)-J_3)}^T& 0\end{array}\right].$$

Moreover,

$$det\left(A_{\theta }^{\prime }\right)={(-1)}^{3\times 3}det\left(B_3\left(Q^{bc}\right)-J_3\right)det\left({(B_3\left(Q^{bc}\right)-J_3)}^T\right)=-{(det(J_3-B_3\left(Q^{bc}\right)))}^2.$$

By exhaustive search, 174 non-degenerate 0–1 matrices of order 3 exist. Suppose two 0–1 matrices of order 3 are either permutationally equivalent or transposes of one another, and they function as bipartite adjacency matrices of two balanced bipartite graphs. In this case, they correspond to isomorphisms of balanced bipartite graphs. Consequently, it is sufficient to consider the following seven 0–1 matrices $B_1,B_2,\dots ,B_7$ (which are listed in ascending order according to the number of edges in $G_k$, where $G_k$ denotes the balanced bipartite graph taking the matrix $B_k$ as its biadjacency matrix). Any of 174 non-degenerate 0–1 matrices is permutationally equivalent to one of the seven matrices or their transposed matrices.

$$B_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],B_2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],B_3=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 1& 1\end{array}\right],B_4=\left[\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right],$$

$$B_5=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right],B_6=\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right],B_7=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 1\\ 0& 1& 1\end{array}\right].$$

In [19], it is proved by the authors that $B_k$ corresponds to the induced subgraph $Q_k$ in $K_{m,n}-G$, where $Q_1\cong 3K_2$, $Q_2\cong K_2\cup P_4$, $Q_3$, $Q_4\cong P_6$, $Q_5\cong C_6$, $Q_6$, or $Q_7$ (see Figure 1, Figure 2 and Figure 3). By Lemma 1 or simple calculations, we know that ${(det(J_3-B\left(Q_k^{bc}\right)))}^2=1$ for $k=1,2,3,4,6,7$ and ${(det(J_3-B\left(Q_5^{bc}\right)))}^2=4$. Hence, the contribution of this case to Equation (7) is given by

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{Q\in {\Omega }_{\{Q_1,Q_2,Q_3,Q_4,Q_6,Q_7\}}(K_{m,n}-G)}}{\displaystyle \frac{-1}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_Q\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_Q\left(v\right))t+\lambda )}}\hfill \\ +\hfill & {\displaystyle \sum _{Q\in {\Omega }_{C_6}(K_{m,n}-G)}}{\displaystyle \frac{-4}{{\displaystyle \prod _{v\in V\left(Q\right)\cap X}}((n-d_Q\left(v\right))t+\lambda ){\displaystyle \prod _{v\in V\left(Q\right)\cap Y}}((m-d_Q\left(v\right))t+\lambda )}}.\hfill \end{array}$$

The proof is completed. □

Remark 2.

Let us assume that $G_1$ and $G_2$ are two graphs sharing a relatively small adjacency rank. When we compute their generalized characteristic polynomials, denoted as $\varphi \left(G_1\right)$ and $\varphi \left(G_2\right)$, any disparity in the number and specific forms of certain substructures between $G_1$ and $G_2$ allows us to promptly infer that $\varphi \left(G_1\right)\ne \varphi \left(G_2\right)$. In a certain sense, Theorem 2 offers us a means to construct cospectral graphs. These graphs, even though they are non-isomorphic, can possess the same adjacency spectra, Laplacian spectra, and possibly other spectral features.

5. Conclusions

In this paper, we studied the computation of the generalized characteristic polynomial or equivalently the zeta function of graphs, and derived a general formula for the generalized characteristic polynomial of the $K_{m,n}$-complement of a bipartite graph. As a by-product, we obtained an explicit formula for the generalized characteristic polynomial of the $K_{m,n}$-complement of a bipartite graph with rank no more than 4. In a sense, the formulas obtained in this paper are straightforward and only rely on the use of fundamental linear algebra about the biadjacency matrix of the bipartite graph.

Future discussions on the generalized characteristic polynomial could focus on the following aspects: In spectral graph theory, its roots (eigenvalues) are the eigenvalues of the graph’s adjacency matrix or other related matrices and expose graph properties, like how the spectral radius relates to connectivity. For graph structure, its coefficients are graph invariants, which contain much information about the graph’s structure, and these invariants are independent of the graph’s labeling and can be used to distinguish different graphs. In graph dynamics, it aids in analyzing system stability during processes like disease spread. In network design, it optimizes network structure by controlling eigenvalues for better connectivity, bandwidth, and fault tolerance performance.