On the spectral characterization of some unicyclic graphs

Let H(n; q, n1, n2) be a graph with n vertices containing a cycle Cq and two hanging paths Pn1 and Pn2 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, n1, n2) are A-cospectral. It is proved that all graphs H(n; q, n1, n2) are determined by their L-spectra. And all graphs H(n; q, n1, n2) are proved to be determined by theirQ -spectra, except for graphsH(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. © 2011 Elsevier B.V. All rights reserved.