Graphs with maximal signless Laplacian spectral radius

Abstract

By the signless Laplacian of a (simple) graph G 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 of G . 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 δ r > δ r - 1 > ⋯ > δ 1 , 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 , l with l + 2 ⩽ i ⩽ r 2 such that δ i + δ r + 1 - i ⩽ n + 1 (respectively, δ i + δ r + 1 - i ⩽ δ l + δ r - 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.