On the spectral radius of quasi-k-cyclic graphs

A connected graph G=(VG,EG) is called a quasi-k-cyclic graph, if there exists a vertex q∈VG 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.