Let be a simple, undirected graph of order and size with vertex set , edge set , adjacency matrix and vertex degrees . The average degree of the neighbor of vertex is . Let be the diagonal matrix of degrees of . Then is the Laplacian matrix of and the signless Laplacian matrix of . Let denote the index of and the index of . We survey upper bounds on and given in terms of the and , as well as of the numbers of common neighbors of pairs of vertices. It is well known that . We show that many but not all upper bounds on are still valid for .
Published September 2007 , 14 pages
This cahier was revised in April 2008