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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.

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$(x-1)(x+1)^2(x-\frac{1-\sqrt{17}}{2})(x-\frac{1+\sqrt{17}}{2})$.

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