Ordering Graphs with Given Size by Their Signless Laplacian Spectral Radii

Let q(G) denote the signless Laplacian spectral radius of a graph G. In this paper, we first give an upper bound on q(G) of a connected graph G with fixed size m >= 3k(k is an element of Z(+)) and maximum degree Delta <= m - k. For two connected graphs G(1) and G(2) with size m >= 4, employing this upper bound, we prove that q(G(1)) > q(G(2)) if Delta(G(1)) > Delta(G(2)) and Delta(G(1)) >= 2m/3 + 1. As an application, we determine the first left perpendiculard/2right perpendicular graphs with the largest signless Laplacian spectral radius among all graphs with fixed size and diameter.