On the spectral characterization of some unicyclic graphs (Q409334)

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 work NEWLINE{\parindent=8mmNEWLINE\begin{itemize}\item[(i)]it is proved 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; NEWLINE\item[(ii)]it is proved that all graphs \(H(n;q,n_1,n_2)\) are determined by their \(L\)-spectra and NEWLINE\item[(iii)]all graphs \(H(n;q,n_1,n_2)\) are determined by their \(Q\)-spectra, except for graphs \(H(2a+4;a+3,\frac{a}{2},\frac{a}{2}+1)\) with \(a\) being a positive even number and \(H(2b;b,\frac{b}{2},\frac{b}{2})\) with \(b\geq 4\) being an even number.NEWLINENEWLINE\end{itemize}}