JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/66210296621029Research ArticleComputing the Permanent of the Laplacian Matrices of Nonbipartite Graphshttps://orcid.org/0000-0001-7848-6870HuXiaoxueKalasoGraceBalibreaFranciscoSchool of ScienceZhejiang University of Science and TechnologyHangzhou 310023Chinazust.edu.cn20212362021202131122020126202123620212021Copyright © 2021 Xiaoxue Hu and Grace Kalaso.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let G be a graph with Laplacian matrix LG. Denote by per LG the permanent of LG. 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 per LG are deduced.

Natural Science Foundation of Zhejiang ProvinceLY20A010005National Natural Science Foundation of China1180151211901525
1. Introduction

All graphs in this paper are restricted to be simple. Let G=VG,EG be a connected simple graph with vertex set VG=1,2,,n and edge set EG. Denoted by dGv, or short for dv if there is no confusion, the degree of vVG. Let G be a graph of order n. The adjacency matrix A=AG=aijn×n related to G is defined as aij=1 if and only if i and j are adjacent and 0 otherwise. Let D=DG be the diagonal matrix of the graph G whose i,i entry is di,i=1,2,,n. The Laplacian matrix related to G is defined by LG=DGAG, 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 , and the references therein.

The determinant and permanent of an n×n square matrix A=aijn×n is defined by(1)det  A=sgnσi=1nai,σi,per A=i=1nai,σi,respectively, where the summation extends over all permutations σ of 1,2,,n and sgnσ=1 if σ is the product of an even number of transpositions and sgnσ=1 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 #P-complete as shown by Valiant . 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  and the references therein for more information on this question.

There are many results on the permanent, or the permanent polynomial perIλA, 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.  first studied the permanent of the Laplacian matrix, in which the polynomial perxILG is suggested to distinguish nonisomorphic trees, and lower bounds on the permanent of LG, per LG, were conjectured. After that, several lower bounds on per LG were proved by Brualdi and Goldwasser  and Merris . For more results on the permanent of the Laplacian matrix, we refer the reader to  and the references contained therein.

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

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

For a given matrix A, can we find two matrices B and C related to A such that per A is the composite of detB and detC?

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 per LGσ and per LiG in terms of Sachs subgraphs, where Gσ is an oriented graph defined in below and LiG=DGAGi with i2=1. 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 per LG are obtained, which can simplify the calculation on the permanent of the Laplacian matrix of more general classes of nonbipartite graphs.

2. Preliminary

Let G be a connected graph with vertex set V and edge set E. An edge eE is called a bridge if the resultant graph obtained from G by deleting the edge e has two components. For a nonempty subset W of VG, the subgraph with vertex set W and edge set consisting of those pairs of vertices that are edges in G is called the induced subgraph of G, denoted by GW. Denote by G\U, where UV, and the graph is obtained from G by removing the vertices of U together with all edges incident to them. Let LiG=DGAGi, where DG and AG are defined as above and i2=1.

Let G be a graph with vertex set V=1,2,,n. The matrix M=mijn×n is called a graphical matrix of G if mij0, ij, if and only if i,j is an edge of G. Then, the adjacency matrix AG, the Laplacian matrix LG, and the matrix LiG are all graphical matrices with respect to the given graph G. Let G be a graph and M be a graphical matrix of G. Suppose that e=i,j is an edge of G and G1 is a subgraph of G. Denote by MG\e the matrix obtained from M by replacing the entries mij and mji by zeros, and by MG1 the principle submatrix of M corresponding to the subgraph G1. Then, MG\e (resp. MG1) is a graphical matrix of G\e (resp. G1) if and only if M is a graphical matrix of G.

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

An oriented graph Gσ is a graph obtained from an undirected graph G by orienting each edge of G a direction. Then, G is referred as the underlying graph of Gσ. 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 C in Gσ is called oddly oriented (resp. evenly oriented) if, for either choice of direction of traversal around C, the number of edges of C 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 Gσ is Pfaffian if every even oriented cycle of G is oddly oriented in Gσ. A graph is called Pfaffian if such a graph has a Pfaffian orientation, see .

Let Gσ be an oriented graph with vertex set VGσ=1,2,,n. The adjacency matrix AGσ=aijn×n of Gσ is defined as aij=aji=1 if ij is an edge of Gσ with tail i and head j and aij=0, otherwise. The Laplacian matrix LGσ of Gσ is defined as LGσ=DGσAGσ, where DGσ is the degree diagonal matrix of its corresponding underlying graph G. Obviously, LGσ is a graphical matrix of Gσ, as well as of G. We refer to [17, 18] and the references therein for more spectral properties on the adjacency matrix of oriented graphs.

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

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

Theorem 1.

Let Gσ be a graph of order n and LGσ be its Laplacian matrix. Then,(2)detLGσ=K1,2,,nHσHKσ1c+Hσ2cHσiKdi,where the first summation is over all induced subgraphs Kσ of Gσ, the second summation is over all Sachs spanning subgraph Hσ of Kσ, and c+Hσ and cHσ are defined as above.

Proof.

Note that LGσ=DGσAGσ; then, by the Laplace expansion formula,(3)detLGσ=K1,2,,n1KdetAKdetGσ\K,where the summation is over all induced subgraphs Kσ of Gσ. Similar to the proof in Theorem 2.3 of ,(4)1KdetAK=HσHKσ1c+Hσ2cHσ,detGσ\K=iKdi,where the summation is over all Sachs spanning subgraph Hσ of Kσ and c+Hσ and cHσ are defined as above. Consequently, the proof is complete.

Denote by ωH the number of components contained in H. Similarly, we can obtain the following result.

Theorem 2.

Let G be a graph of order n and LiG=DGAGi. Then,(5)detLiG=K1,2,,niKHHK1ωH2cHiKdi,where the first summation is over all subgraphs K of G, the second summation is over all Sach spanning subgraph H of K, and cH denotes the number of cycles contained in H.

3. The Permanent of the Laplacian Matrix of a Graph

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

Lemma 1 (see Lemma 2.1 in [<xref ref-type="bibr" rid="B13">13</xref>]).

Let G be a graph of order n and LG be its Laplacian matrix. Then,(6)per LG=K1,2,,nHHK1coH2cHiKdi,where the first summation is over all induced subgraphs K of G, the second summation is over all Sachs spanning subgraph H of K, and coH and cH denote the number of odd cycles and cycles contained in H, respectively.

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

Theorem 3.

Let G be a bipartite Pfaffian graph of order n and let σ be an orientation of G such that Gσ is a Pfaffian oriented graph. Let LG and LGσ be the Laplacian matrices of G and Gσ, respectively. Then,(7)per LG=detLGσ.

Proof.

Let H be any Sachs subgraph of G and the corresponding Sachs oriented subgraph of Gσ be denoted by Hσ. Since G is bipartite, the order of H is even and CoH=0. Thus,(8)1coH2cHiHdi=2cHiHdi.

Since Gσ is Pfaffian, then Hσ is Pfaffian and c+Hσ=0. Thus,(9)1c+Hσ2ceHσiHσdi=2ceHσiHσdi,where ceHσ denotes the number of all even cycles contained in Hσ. 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 LG.

Let G1 be the set of all Pfaffian graphs in which each element G of G1 satisfies the following:

Each odd cycle contained in G has length 3mod  4

G contains no disjoint odd cycles, that is, two arbitrary odd cycles have at least one common vertex

For any odd cycle C, GC contains no cycles with length 0mod  4

Theorem 4.

Let G be a graph with uVG. Then,(10)per LG=per LG\udGu+eincident to uper LG\e+2uC1Cper LG\C,where the first summation is over all edges incident to u and the second summation is over all cycle C containing the vertex u.

Proof.

All Sachs subgraphs of G can be divided into three kinds: those that contain the edge e, incident to the vertex u, as a single edge, those that contain u as a vertex of a cycle, and those that do not. One finds that the sum of all summands of the former is pereLG\e, the sum of all summands of the second kind is 2uC1Cper LG\C, and the sum of all summands of the third kind is per LG\udGu. Thus, the result is as follows.

Theorem 5.

Let G be a graph with e=u,vEG. Then,(11)per LG=per LG\u,v+per LG\e+2eC1Cper LG\C,where the summation is over all cycle C containing the edge e.

Proof.

All Sachs subgraphs of G can be divided into three kinds: those that contain the edge e as a single edge, those that contain e as an edge of a cycle, and those that do not. One finds that the sum of all summands of the former is per LG\u,v, the sum of all summands of the second kind is 2eC1Cper LG\C, and the sum of all summands of the third kind is per LG\e. Thus, the result is as follows..

As a consequence of Theorem 5, we have the following result.

Corollary 1.

Let G1 and G2 be two disjoint graphs with uG1 and vG2, and G be the graph obtained from G1 and G2 by adding an edge between u and v. Then,(12)per LG=per LG1per LG2+per LG1\uper LG2\v.

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).

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