Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs

Abstract

Let be a graph with Laplacian matrix . Denote by the permanent of . In this study, we investigate the problem of computing the permanent of the Laplacian matrix of nonbipartite graphs. We show that the permanent of the Laplacian matrix of some classes of nonbipartite graphs can be formulated as the composite of the determinants of two matrices related to those Laplacian matrices. In addition, some recursion formulas on are deduced.

1. Introduction

All graphs in this paper are restricted to be simple. Let be a connected simple graph with vertex set and edge set . Denoted by , or short for if there is no confusion, the degree of . Let be a graph of order . The adjacency matrix related to is defined as if and only if and are adjacent and 0 otherwise. Let be the diagonal matrix of the graph whose entry is . The Laplacian matrix related to is defined by , which has been extensively investigated for a long time. For more properties of the Laplacian matrix of graphs, reader may refer to the books [1, 2], the surveys [3–5], and the references therein.

The determinant and permanent of an square matrix is defined byrespectively, where the summation extends over all permutations of and if is the product of an even number of transpositions and otherwise.

The formula for permanent is similar to, even simpler than, the formula for determinant. However, there is a polynomial algorithm for calculating determinants, whereas calculating permanents is -complete as shown by Valiant [6]. Therefore, it is reasonable to ask if perhaps computing the permanent of a matrix can be somehow converted to computing the determinant of a related matrix. Readers may refer to [7–9] and the references therein for more information on this question.

There are many results on the permanent, or the permanent polynomial , in terms of the adjacency matrix of bipartite graphs, see, for example, [10, 11], whereas, there are few formulaic results on the permanent of the matrix related to the nonbipartite graphs.

Merris et al. [12] first studied the permanent of the Laplacian matrix, in which the polynomial is suggested to distinguish nonisomorphic trees, and lower bounds on the permanent of , , were conjectured. After that, several lower bounds on were proved by Brualdi and Goldwasser [13] and Merris [11]. For more results on the permanent of the Laplacian matrix, we refer the reader to [11–15] and the references contained therein.

Pólya’s permanent problem is well known, which has many equivalent versions, such as, given a matrix , it was possible to change the signs of some of the entries of to give a new matrix such that the corresponding terms of and were equal. Plya’s permanent problem still remains open. We refer the readers to the survey [16] for the history and different versions of Plya’s permanent problem.

In general, for Plya’s permanent problem, we only consider exactly one objective matrix, that is, for a given matrix , we want to find one objective matrix related to such that . Naturally, the following question, which can be considered as a generalized version Plya’s permanent problem, is interesting.

For a given matrix , can we find two matrices and related to such that is the composite of and ?

We, in this paper, continue to investigate the problem of computing the permanent of the Laplacian matrix of graphs. In Section 2, we give some preliminary results, including the combinatorial description of and in terms of Sachs subgraphs, where is an oriented graph defined in below and with . In Section 3, we first deduce a formula on the permanent of the Laplacian matrix of a class of bipartite graphs. Then, we show that the permanent of the Laplacian matrix of a class of nonbipartite graphs can be formulated as the composite of the determinants of two matrices related to their Laplacian matrices. In addition, some recursion formulas on are obtained, which can simplify the calculation on the permanent of the Laplacian matrix of more general classes of nonbipartite graphs.

2. Preliminary

Let be a connected graph with vertex set and edge set . An edge is called a bridge if the resultant graph obtained from by deleting the edge has two components. For a nonempty subset of , the subgraph with vertex set and edge set consisting of those pairs of vertices that are edges in is called the induced subgraph of , denoted by . Denote by , where , and the graph is obtained from by removing the vertices of together with all edges incident to them. Let , where and are defined as above and .

Let be a graph with vertex set . The matrix is called a graphical matrix of if , , if and only if is an edge of . Then, the adjacency matrix , the Laplacian matrix , and the matrix are all graphical matrices with respect to the given graph . Let be a graph and be a graphical matrix of . Suppose that is an edge of and is a subgraph of . Denote by the matrix obtained from by replacing the entries and by zeros, and by the principle submatrix of corresponding to the subgraph . Then, (resp. ) is a graphical matrix of (resp. ) if and only if is a graphical matrix of .

For an undirected graph , the subgraph of is called a Sachs subgraph of if each component of is either a single edge or a cycle; see, for instance, [16]. For a Sachs subgraph , denote, by and , the number of odd cycles and cycles contained in , respectively.

An oriented graph is a graph obtained from an undirected graph by orienting each edge of a direction. Then, is referred as the underlying graph of . We should point out that our oriented graph considered its underlying graph in terms of defining matching, degree, path, and connectedness. An even oriented cycle in is called oddly oriented (resp. evenly oriented) if, for either choice of direction of traversal around , the number of edges of directed in the direction of traversal is odd (resp. even). Clearly, this is independent of the initial choice of direction of traversal. An oriented graph is Pfaffian if every even oriented cycle of is oddly oriented in . A graph is called Pfaffian if such a graph has a Pfaffian orientation, see [16].

Let be an oriented graph with vertex set . The adjacency matrix of is defined as if is an edge of with tail and head and , otherwise. The Laplacian matrix of is defined as , where is the degree diagonal matrix of its corresponding underlying graph . Obviously, is a graphical matrix of , as well as of . We refer to [17, 18] and the references therein for more spectral properties on the adjacency matrix of oriented graphs.

The subgraph, denoted by , of a given oriented graph is called a Sachs oriented subgraph of if each component of is either a single edge or a cycle with length even; see examples [18]. For a given Sachs oriented subgraph , denote by and the number of evenly even cycles and cycles contained in , respectively.

For a given oriented graph , an argument similar to the one given in the proof of Theorem 2.1 in pp. 276–277 of [11] yields the combinatorial description of ; for completeness, we give a simple proof.

Theorem 1. Let be a graph of order and be its Laplacian matrix. Then,where the first summation is over all induced subgraphs of , the second summation is over all Sachs spanning subgraph of , and and are defined as above.

Proof. Note that ; then, by the Laplace expansion formula,where the summation is over all induced subgraphs of . Similar to the proof in Theorem 2.3 of [18],where the summation is over all Sachs spanning subgraph of and and are defined as above. Consequently, the proof is complete.
Denote by the number of components contained in . Similarly, we can obtain the following result.

Theorem 2. Let be a graph of order and . Then,where the first summation is over all subgraphs of , the second summation is over all Sach spanning subgraph of , and denotes the number of cycles contained in .

3. The Permanent of the Laplacian Matrix of a Graph

From the work of Brualdi and Goldwasser [13], a formula on the permanent of the Laplacian matrix of a graph is given as follows.

Lemma 1 (see Lemma 2.1 in [13]). Let be a graph of order and be its Laplacian matrix. Then,where the first summation is over all induced subgraphs of , the second summation is over all Sachs spanning subgraph of , and and denote the number of odd cycles and cycles contained in , respectively.

Combining with Theorem 1 and Lemma 1, we have the following theorem.

Theorem 3. Let be a bipartite Pfaffian graph of order and let be an orientation of such that is a Pfaffian oriented graph. Let and be the Laplacian matrices of and , respectively. Then,

Proof. Let be any Sachs subgraph of and the corresponding Sachs oriented subgraph of be denoted by . Since is bipartite, the order of is even and . Thus,Since is Pfaffian, then is Pfaffian and . Thus,where denotes the number of all even cycles contained in . Consequently, the result is as follows.
Theorem 3 is invalid to graphs containing odd cycles. Roughly speaking, any odd cycle has no contribution to the determinant of the Laplacian matrix of an oriented graph. However, for the permanent of the Laplacian matrix of an undirected graph, the effect of odd cycles is completely different. Henceforth, it is difficulty to compute the permanent of the Laplacian matrix of a nonbipartite graph.
In the following, we will show that there exists a class of nonpartite graphs such that the permanent of the Laplacian matrix of those graphs can be formulated as the composite of the determinants of two matrices related to .
Let be the set of all Pfaffian graphs in which each element of satisfies the following:(1)Each odd cycle contained in has length (2) contains no disjoint odd cycles, that is, two arbitrary odd cycles have at least one common vertex(3)For any odd cycle , contains no cycles with length

Theorem 4. Let be a graph with . Then,where the first summation is over all edges incident to and the second summation is over all cycle containing the vertex .

Proof. All Sachs subgraphs of can be divided into three kinds: those that contain the edge , incident to the vertex , as a single edge, those that contain as a vertex of a cycle, and those that do not. One finds that the sum of all summands of the former is , the sum of all summands of the second kind is , and the sum of all summands of the third kind is . Thus, the result is as follows.

Theorem 5. Let be a graph with . Then,where the summation is over all cycle containing the edge .

Proof. All Sachs subgraphs of can be divided into three kinds: those that contain the edge as a single edge, those that contain as an edge of a cycle, and those that do not. One finds that the sum of all summands of the former is , the sum of all summands of the second kind is , and the sum of all summands of the third kind is . Thus, the result is as follows..
As a consequence of Theorem 5, we have the following result.

Corollary 1. Let and be two disjoint graphs with and , and be the graph obtained from and by adding an edge between and . Then,

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by Zhejiang Provincial Natural Science Foundation of China (no. LY20A010005) and National Natural Science Foundation of China (nos. 11801512 and 11901525).