I want to find all eigenvalues of the adjacency matrix of the following graph(Graph spectrum), where $G$ and $H$ are complete graphs with $n$ and $m$ vertices, respectively, for positive integers $n,m \ge 3$.
Using Interlacing Theorem and by deleting the vertices $x$ and $y$, I can find some eigenvalues that are equal to $-1$. Also by Schwenk Theorem, I can find the characteristic polynomial of this graph, but finding the roots of such polynomial is not simple. Furthermore, we can use $\Sigma_{i=1}^{m+n+2}{\lambda_i^{k}}$ that is equal to the number of closed walks of length $k$ in this graph, but I could not find all eigenvalues with this method because the resulted system of equations is too hard to solve without using some softwares.
I am looking for some simple methods for finding the spectrum of such graphs, but I don't know it is possible or not.
Thanks in advance