A note on (signless) Laplacian spectral ordering with maximum degrees of graphs

An (n,m)-graph is referred to be a graph with n vertices and m edges. Let Δ(G) and δ(G) be the maximum and minimum degree of a graph G and let μ(G) and q(G) be the Laplacian and signless Laplacian spectral radius of G, respectively. In this paper, we prove that for two connected nonregular (n,m)-graphs G and G′, if Δ(G)≥2m−(n−1)δ(G′)δ(G′)+1+δ(G′) and Δ(G)>Δ(G′)+δ(G′)−1, then μ(G)>μ(G′) and q(G)>q(G′). Also, we obtain that for two connected nonregular (n,m)-graphs G and G′, if Δ(G)≥2m−(n−1)δ(G′)δ(G′)+1+1 and Δ(G)>Δ(G′), then μ(G)>μ(G′)−δ(G′)+1 and q(G)>q(G′)−δ(G′)+1.