By the signless Laplacian of a (simple) graphG we mean the matrix Q(G) = D(G)+A(G), where A(G), D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees ofG. It is known that connected graphs G that maximize the signless Laplacian spectral radius ρ(Q(G)) over all connected graphs with given numbers of vertices and edges are (degree) maximal. For a maximal graph G with n vertices and r distinct vertex degrees δr > δr−1 >··· > δ1, it is proved that ρ(Q(G)) < ρ(Q(H))for some maximal graph H with n+1 (respectively, n) vertices and the same number of edges as G if either G has precisely two dominating vertices or there exists an integer i,2 ≦ i ≦ [r/2] (respectively, if there exist positive integers i, lwithl + 2 ≦ i ≦ [r/2] such that δi + δr+1−i ≦ n+1 (respectively, δi + δr+1−i ≦ δl + δr−l + 1). Graphs that maximize ρ(Q(G)) over the class of graphs with m edges and m−k vertices, for k = 0,1,2,3, are completely determined.
Relation:
Linear Algebra and its Applications 432(7), pp.1708–1733
