Computing the Permanent of the Laplacian Matrices of Nonbipartite Graphs

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