The characterizing properties of (signless) Laplacian permanental polynomials of bicyclic graphs

Abstract

Let $G$ be a graph with $n$ vertices, and let $L(G)$ and $Q(G)$ be the Laplacian matrix and signless Laplacian matrix of $G$, respectively. The polynomial $\pi (L(G);x)=\mathrm{per}(xI-L(G))$ (resp. $\pi (Q(G);x)=\mathrm{per}(xI-Q(G))$) is called {\em Laplacian permanental polynomial} (resp. {\em signless Laplacian permanental polynomial}) of $G$. In this paper, we show that two classes of bicyclic graphs are determined by their (signless) Laplacian permanental polynomials.