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

TL;DR

This paper investigates the properties of Laplacian and signless Laplacian permanental polynomials in bicyclic graphs, demonstrating that certain classes of these graphs are uniquely identified by these polynomials.

Contribution

It establishes that two classes of bicyclic graphs can be uniquely characterized by their (signless) Laplacian permanental polynomials, a novel identification result.

Findings

Two classes of bicyclic graphs are determined by their (signless) Laplacian permanental polynomials.

The paper provides methods to distinguish these graphs using polynomial invariants.

Results contribute to graph characterization via algebraic invariants.

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 $\pi(L(G);x)={\rm per}(xI-L(G))$ (resp. $\pi(Q(G);x)={\rm per}(xI-Q(G))$) 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.