DistancePolynomial returns the distance polynomial for the connected graph G in the variable x.

For an undirected connected graph G, the distance polynomial of G is defined to be the characteristic polynomial of the graph distance matrix.

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

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

$P≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 3\; edge(s)}$

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

$x^4-20x^2-32x-12$

$C≔\mathrm{CycleGraph}\left(5\right)$

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

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

$x^5-25x^3-60x^2-35x-6$

The GraphTheory[DistancePolynomial] command was introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.