Spectrum of adjacency matrix of block graph Let us consider a graph $G$ having $m$ number of complete sub-graphs $K_{n_1},K_{n_2},...,K_{n_m}$ which have size $n_1,n_2,...,n_m$ respectively. Further $\forall i$, one vertex of $K_{n_i}$ is connected to one vertex of $K_{n_{i+1}}$ by an edge. Similarly, one vertex (different from previous one) of $K_{n_i}$ is connected to one vertex of $K_{n_{i-1}}$ by an edge. In this way, complete sub-graphs are connected in chain to form $G$. Find the eigenvalues of adjacency matrix of $G$.
 A: The characteristic polynomial of a complete graph with $n$ vertices is $(x-n+1)(x+1)^{n-1}$. By deleting a vertex from the complete graph $K_n$, the remaining graph is the complete graph $K_{n-1}$ with characteristic polynomial $(x-n+2)(x+1)^{n-2}$.
Consider subgraph $G_i$ of the graph $G$ that is constructed from the union of two complete subgraphs $K_{n_i}$ and $K_{n_{i+1}}$ of $G$ and the edge which connect these two subgraphs. By using the following theorem of Schwenk, you can find the characteristic polynomial of $G_i$.
Schwenk A. J., Computing the characteristic polynomial of a graph, in Graphs 
and Combinatorics (eds. Bari, R., Harary, F.), Springer-Verlag (New York), 1974. 


For any edge $uv$ of the graph $G$,
    \begin{align*}
P_G(x)=P_{G-uv}(x)-P_{G-u-v}(x)-2\sum_{Z\in C(uv)}P_{G-V(Z)}(x),
\end{align*}
    where $C(uv)$ denotes the set of all cycles containing $uv$.


Since the edge which connect the two complete subgraphs in $G_i$ is not contained in any cycles, the third part of the above formula is zero. So, the characteristic polynomial of $G_i$ is
\begin{align*}
(x-n_i+1)(x+1)^{n_i-1}(x-n_{i+1}+1)(x+1)^{n_{i+1}-1}\\
-(x-n_i+2)(x+1)^{n_i-2}(x-n_{i+1}+2)(x+1)^{n_{i+1}-2}
\end{align*}
Such as above and by using the above theorem repeatedly, you can find the characteristic polynomial of $G$. 
As Meyerowitz mentioned, you can find some roots of this characteristic polynomial that is equal to $-1$ simply, but finding some of them is difficult in general. 
