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

$G≔\mathrm{GraphComplement}\left(\mathrm{CompleteGraph}\left(3\,4\right)\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 7\; vertices\; and\; 9\; edge(s)}$

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

$\mathrm{FindClique}\left(G\,3\right)$

$\left[1\,2\,3\right]$

$\mathrm{FindClique}\left(G\,4\right)$

$\left[4\,5\,6\,7\right]$

$\mathrm{G1}≔\mathrm{Graph}\left(5\,\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{1\,4\right\}\,\left\{1\,5\right\}\right\}\right)$

$\mathrm{G1}≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 4\; edge(s)}$

$\mathrm{G2}≔\mathrm{Graph}\left(5\,\left\{\left\{1\,2\right\}\,\left\{1\,3\right\}\,\left\{1\,4\right\}\,\left\{1\,5\right\}\,\left\{2\,3\right\}\,\left\{2\,5\right\}\,\left\{3\,4\right\}\,\left\{4\,5\right\}\right\}\right)$

$\mathrm{G2}≔\mathrm{Graph\; 3:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 8\; edge(s)}$

$\mathrm{DrawGraph}\left(\mathrm{G1}\right)$

$\mathrm{DrawGraph}\left(\mathrm{G2}\right)$

$\mathrm{DrawGraph}\left(\mathrm{GraphIntersection}\left(\mathrm{G1}\,\mathrm{G2}\right)\right)$

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

$P≔\mathrm{Graph}\left(\left\{\left\{1\,2\right\}\,\left\{2\,3\right\}\,\left\{3\,4\right\}\right\}\right)$

$P≔\mathrm{Graph\; 4:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

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

$3x^2+4x+1$

$C≔\mathrm{CycleGraph}\left(5\right)$

$C≔\mathrm{Graph\; 5:\; an\; undirected\; unweighted\; graph\; with\; 5\; vertices\; and\; 5\; edge(s)}$

$\mathrm{IndependencePolynomial}\left(C\,x\right)$

$5x^2+5x+1$

$M≔\mathrm{MirzakhaniGraph}\left(\right)$

$M≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 63\; vertices\; and\; 183\; edge(s)}$

$\mathrm{ChromaticNumber}\left(M\,\mathrm{method}=\mathrm{sat}\right)$

$3$