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

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 ϕ ( Q ( G ) , x ) = per ( x I n - Q ( G ) ) and ϕ ( L ( G ) , x ) = per ( x I 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 ϕ ( Q ( G ) , x ) . We obtain the real root distribution of ϕ ( 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 ϕ ( Q ( G ) , x ) . And we also determine the relation between the multiplicity of the root p of ϕ ( 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.