Elsevier, Linear Algebra and its Applications, 8-10(433), p. 1561-1572, 2010
DOI: 10.1016/j.laa.2010.06.007
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A connected graph G = (V(G), E(G)) is called a quasi-k-cyclic graph, if there exists a vertex q is an element of 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.