Some results on signless Laplacian coefficients of graphs

Abstract

Let QG(x)=det(xI-Q(G))=∑i=0n(-1)iζixn-i be the characteristic polynomial of the signless Laplacian matrix of a graph G. Due to the nice properties of the signless Laplacian matrix, Q(G), in comparison with the other matrices related to graphs, ζ-ordering, an ordering based on the coefficients of the signless Laplacian characteristic polynomial, is of interest. In this paper, using graph transformations, we establish some relations between the graph structure and its coefficients of the signless Laplacian characteristic polynomial. So, we express some results about ζ-ordering of graphs, focusing our attention to the unicyclic graphs. Finally, as an application of these results, we discuss the ordering of graphs based on their incidence energy.