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 n 1 and P n 2 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 ( 2 a + 4 ; a + 3 , a 2 , a 2 + 1 ) with a being a positive even number and H ( 2 b ; b , b 2 , b 2 ) with b ≥ 4 being an even number. Moreover, the Q -cospectral graphs with these two exceptions are given.