Recursive formulae for the Laplacian permanental polynomials of signed graphs

A signed graph [Formula: see text] is defined on a graph [Formula: see text] (known as the underlying graph of [Formula: see text]), where the edges are assigned [Formula: see text] or − sign by a sign function [Formula: see text]. In this paper, we introduce the Laplacian permanental polynomial of a signed graph. The matrix [Formula: see text] is defined as the Laplacian matrix of [Formula: see text], where [Formula: see text] is the degree diagonal matrix of the underlying graph [Formula: see text] and [Formula: see text] is the adjacency matrix of [Formula: see text]. The polynomial [Formula: see text] is called the Laplacian permanental polynomial of [Formula: see text], where [Formula: see text] is the identity matrix and [Formula: see text] denotes the permanent of a matrix. In this paper, we develop some recursive formulae for computing [Formula: see text]. We study the polynomial [Formula: see text], when [Formula: see text] is a balanced signed graph, a signed tree, a balanced signed bipartite graph, and a signed graph with only odd-length negative cycles. Moreover, for any signed graph [Formula: see text], we compute the general expressions for the first five coefficients and the constant term of [Formula: see text].