Graphs whose signless Laplacian spectral radius does not exceed the Hoffman limit value

Abstract: For a graph matrix M, the Hoffman limit value is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of , where the graph is obtained by attaching a pendant edge to the cycle of length . In spectral graph theory, M is usually either the adjacency matrix A or the Laplacian matrix L or the signless Laplacian matrix Q. The exact values of and were first determined by Hoffman and Guo, respectively. Since is bipartite for odd n, we have . All graphs whose A-index is not greater than were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed . The results obtained are determinant to describe all graphs whose L-index is not greater then . This is done precisely in Wang et al. (in press) . [Copyright &y& Elsevier]