IsIntegerGraph returns true if its argument G is an integer graph, namely if the spectrum of G consists of only integer numbers, and false otherwise.

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

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

$\mathrm{K3}≔\mathrm{CompleteGraph}\left(3\right)$

$\mathrm{K3}≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 3\; vertices\; and\; 3\; edge(s)}$

$\mathrm{IsIntegerGraph}\left(\mathrm{K3}\right)$

$\mathrm{true}$

$\mathrm{factor}\left(\mathrm{CharacteristicPolynomial}\left(\mathrm{K3}\,x\right)\right)$

$\left(x-2\right){\left(x+1\right)}^2$

$\mathrm{P3}≔\mathrm{PathGraph}\left(3\right)$

$\mathrm{P3}≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 3\; vertices\; and\; 2\; edge(s)}$

$\mathrm{IsIntegerGraph}\left(\mathrm{P3}\right)$

$\mathrm{false}$

$\mathrm{factor}\left(\mathrm{CharacteristicPolynomial}\left(\mathrm{P3}\,x\right)\right)$

$x\left(x^2-2\right)$

$H≔\mathrm{LeviGraph}\left(\right)$

$H≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 30\; vertices\; and\; 45\; edge(s)}$

$\mathrm{IsIntegerGraph}\left(H\right)$

$\mathrm{true}$

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

$x^{10}\left(x-3\right)\left(x+3\right){\left(x-2\right)}^9{\left(x+2\right)}^9$