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

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(Hn), where the graph Hn is obtained by attaching a pendant edge to the cycle Cn-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 Hn 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].