Graphs whose second largest signless Laplacian eigenvalue does not exceed [formula omitted

Abstract

For a graph G, let the signless Laplacian matrix Q(G) defined as Q(G)=D(G)+A(G), where A(G) and D(G) are, respectively, the adjacency matrix and the degree matrix of G. The Q-eigenvalues of G are the eigenvalues of Q(G). In this paper, we characterize the connected graphs whose second largest Q-eigenvalue κ2 does not exceed 2+2, obtain all the minimal forbidden subgraphs with respect to this property, and discover a large family of such graphs that are determined by their Q-spectrum. The connected graphs G such that κ2(G)=2+2 are also detected.