On the Permanental Polynomial and Permanental Sum of Signed Graphs

Abstract Let Ġ = (G, σ) be a signed graph, where G is its underlying graph and σ is its sign function (defined on the edge set E(G) of G). Let A(Ġ) be the adjacency matrix of Ġ. The polynomial π(Ġ, x) = per(xI −A(Ġ)) is called the permanental polynomial of Ġ, where I is the identity matrix and per denotes the permanent of a matrix. In this paper, we obtain the coefficients of the permanental polynomial of a signed graph in terms of its structure. We also establish the recursion formulas for the permanental polynomial of a signed graph. Moreover, we investigate the permanental sum PS(Ġ) of a signed graph Ġ, give the recursion formulas for the permanental sum PS(Ġ), and show that the equation PS(Ġ) = PS(G) holds for trees and unicyclic graphs, where PS(G) is the permanental sum of the underlying graph G of Ġ.