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

Abstract

For a graph matrix M, the Hoffman limit value H ( M ) is the limit (if it exists) of the largest eigenvalue (or, M-index, for short) of M ( H n ) , where the graph H n is obtained by attaching a pendant edge to the cycle C n - 1 of length n - 1 . 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 H ( A ) and H ( L ) were first determined by Hoffman and Guo, respectively. Since H n is bipartite for odd n, we have H ( Q ) = H ( L ) . All graphs whose A-index is not greater than H ( A ) were completely described in the literature. In the present paper, we determine all graphs whose Q-index does not exceed H ( Q ) . The results obtained are determinant to describe all graphs whose L-index is not greater then H ( L ) . This is done precisely in Wang et al. (in press) [21].