Connected graphs with maximal Q-index: The one-dominating-vertex case

By the signless Laplacian of a (simple) graph G we mean the matrix Q(G) = D(G) + A(C). where A(G). D(G) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. For every pair of positive integers n, k, it is proved that if 3 <= k <= n - 3, then H-n,H-k, the graph obtained from the star K-1,K- n-1 by joining a vertex of degree 1 to k+1 other vertices of degree 1, is the unique connected graph that maximizes the largest signless Laplacian eigenvalue over all connected graphs with n vertices and n + k edges. (C) 2011 Elsevier Inc. All rights reserved.