CharacteristicPolynomial returns the characteristic polynomial of the adjacency matrix of a graph G, as a polynomial in x.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{SpecialGraphs}\right)\:$

$P≔\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\right\}\right)$

$P≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 3\; vertices\; and\; 2\; edge(s)}$

$\mathrm{CharacteristicPolynomial}\left(P\,x\right)$

$x^3-2x$

$A≔\mathrm{AdjacencyMatrix}\left(P\right)$

$A≔\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$

$\mathrm{LinearAlgebra}\left[\mathrm{CharacteristicPolynomial}\right]\left(A\,x\right)$

$x^3-2x$

$G≔\mathrm{ShrikhandeGraph}\left(\right)\:$

$\mathrm{Diameter}\left(G\right)$

$2$

$f≔\mathrm{CharacteristicPolynomial}\left(G\,x\right)$

$f≔x^{16}-48x^{14}-64x^{13}+768x^{12}+1536x^{11}-5888x^{10}-15360x^9+23040x^8+81920x^7-36864x^6-245760x^5-32768x^4+393216x^3+196608x^2-262144x-196608$

$\mathrm{eigvals}≔\left\{\mathrm{solve}\left(f\right)\right\}$

$\mathrm{eigvals}≔\left\{\mathrm{-2}\,2\,6\right\}$

$\mathrm{numelems}\left(\mathrm{eigvals}\right)$

$3$