Further results on the star degree of graphs

Abstract

Let G be any simple undirected graph and let Q(G) be the signless Laplacian matrix of G. The polynomial ϕ(Q(G),x)=per(xI−Q(G)) is called the signless Laplacian permanental polynomial of G. The star degree of a graph G is the multiplicity of root 1 of ϕ(Q(G),x). Faria (1985) first considered the star degree of graphs. Based on Faria’s results, we further study the features of star degree of graphs, and give a formula to compute the star degree of a graph by a vertex partition of the graph. As applications, we derive the star degree set of n-vertex graphs, and we determine the graphs with extremal star degree. Furthermore, we show that some graphs with given star degree are determined by their signless Laplacian permanental spectra.