On the spectral radius of quasi-k-cyclic graphs

Abstract A connected graph G = ( V G , E G ) is called a quasi-k-cyclic graph, if there exists a vertex q ∈ V G such that G - q is a k-cyclic graph (connected with cyclomatic number k). In this paper we identify in the set of quasi-k-cyclic graphs (for k ⩽ 3 ) those graphs whose spectral radius of the adjacency matrix (and the signless Laplacian if k ⩽ 2 ) is the largest. In addition, for quasi-unicyclic graphs we identify as well those graphs whose spectral radius of the adjacency matrix is the second largest.