On the spectral characterization of some unicyclic graphs

Let H(n; q, n(1), n(2)) be a graph with n vertices containing a cycle C(q) and two hanging paths P(n1) and P(n2) attached at the same vertex of the cycle. In this paper, we prove that except for the A-cospectral graphs H(12; 6, 1,5) and H(12: 8. 2. 2), no two non-isomorphic graphs of the form H(n; q, n(1), n(2)) are A-cospectral. It is proved that all graphs H(n; q. n(1), n(2)) are determined by their L-spectra. And all graphs H(n; q, n(1), n(2)) are proved to be determined by their Q-spectra, except for graphs H(2a + 4; a + 3, a/2, a/2 + 1) with a being a positive even number and H(2b: b, b/2, b/2) with b >= 4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given. (C) 2011 Elsevier By. All rights reserved