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. For every pair of positive integers n , k , it is proved that if 3 ⩽ k ⩽ n - 3 , then H n , k , the graph obtained from the star K 1 , 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.
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