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 their Q-spectra, except for graphs H(2a+4;a+3,a2,a2+1) with a being a positive even number and H(2b;b,b2,b2) with b≥4 being an even number. Moreover, the Q-cospectral graphs with these two exceptions are given.

► The spectral characterization of the unicyclic graph H(n;q,n1,n2) is investigated. ► Some graphs A-cospectral with H(n;q,n1,n2) are figured out. ► All graphs H(n;q,n1,n2) are determined by their L-spectra. ► Graphs H(n;q,n1,n2), with only two exceptions, are determined by their Q-spectra.