Let G be any simple undirected graph and let Q(G) be the signless Laplacian matrix of G . The polynomial phi(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 phi(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. (c) 2022 Elsevier Inc. All rights reserved.
NSTL
Elsevier
