On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs

Abstract

Let G be a graph, and let Q(G) and L(G) denote the signless Laplacian matrix and the Laplacian matrix of G, respectively. The polynomials $\phi (Q(G),x)={\textrm{per}}(xI_{n}-Q(G))$ and $\phi (L(G),x)={\textrm{per}}(xI_{n}-L(G))$ are called signless Laplacian permanental polynomial and Laplacian permanental polynomial of G, respectively. In this paper, we investigate the properties of roots of $\phi (Q(G),x)$. We obtain the real root distribution of $\phi (Q(G),x)$. In particular, using the Gallai–Edmonds structure theorem, we determine the structures of graphs G whose roots of signless Laplacian permanental polynomial of G contain no positive integer p, where p is the minimum vertex degree of G. And we determine completely the graphs each of which having the multiplicity of the integer root p is equal to the deficiency of a maximum p-pendant structure of the graph. These results extend the conclusion obtained by Faria (Linear Algebra Appl 299:15–35, 1995). Furthermore, we give an algorithm to calculate the multiplicity of the root p of $\phi (Q(G),x)$. And we also determine the relation between the multiplicity of the root p of $\phi (Q(G),x)$ and the matching number of G. Finally, we investigate the properties of roots of Laplacian permanental polynomial of non-bipartite graphs. And some open problems are presented.

Appendix A

The Laplacian permanental polynomials of all non-bipartite graphs with $n(3\le n\le 5)$ vertices and their roots, for details of these graphs see the appendix in [5] (Table 1).