<s>[INST] Find the spectrum of graphs for adjacency matrix Find the spectrum of graphs for adjacency matrix $(A)$ below :
$$ \left[
        \begin{matrix}
        0 & 1 & 1 & 0 & 0 \\
        1 & 0 & 1 & 0 & 0 \\
        1 & 1 & 0 & 1 & 1 \\
        0 & 0 & 1 & 0 & 1 \\
        0 & 0 & 1 & 1 & 0 \\
        \end{matrix} 
\right] $$ 
My attempt :
The spectrum of a graph is the list of eigenvalues of the adjacency matrix with the multiplicities. I do the calculation by $det (xI - A)$, I find the $(xI - A)$ as below :
$$ \left[
        \begin{matrix}
        x  & -1 & -1 & 0  & 0 \\
        -1 & x  & -1 & 0  & 0 \\
        -1 & -1 & x  & -1 & -1 \\
        0  & 0  & -1 & x  & -1 \\
        0  & 0  & -1 & -1 & x \\
        \end{matrix} 
\right] $$ 
$det (xI - A) = x^5-6x^3-4x^2+5x+4$
Then I stack to find the eigenvalues of the adjacency matrix with the multiplicities. 
 [/INST] $(x-1)(x+1)^2(x-\frac{1-\sqrt{17}}{2})(x-\frac{1+\sqrt{17}}{2})$.
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