The AllPairsDistance command returns a square matrix A where $A_{i,j}$ is the distance from vertex i to vertex j in the graph G, that is, the length of the shortest path from vertex i to vertex j.  If G is not a weighted graph, then edges have weight 1. If there is no path, then the distance is infinite and $A_{i,j}=\mathrm{\infty }$.

This procedure is an implementation of the Floyd-Warshall all-pairs shortest path algorithm.  The complexity is $\mathrm{O}\left(n^3\right)$ where n is the number of vertices of G.

To compute distances or shortest paths from a single vertex to every other vertex, use either DijkstrasAlgorithm or BellmanFordAlgorithm because they are more efficient.

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

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

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 5\; vertices\; and\; 8\; edge(s)}$

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

$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ 1& 0& 1& 2& 1\\ 1& 1& 0& 1& 2\\ 1& 2& 1& 0& 1\\ 1& 1& 2& 1& 0\end{array}\right]$

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

$2$

$H≔\mathrm{Graph}\left(\mathrm{directed}\,\left\{\mathrm{seq}\left(\left[1\,i\right]\,i=2..5\right)\right\}\,\mathrm{Trail}\left(2\,3\,4\,5\,2\right)\right)$

$H≔\mathrm{Graph\; 2:\; a\; directed\; graph\; with\; 5\; vertices\; and\; 8\; arc(s)}$

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

$\left[\begin{array}{ccccc}0& 1& 1& 1& 1\\ \mathrm{\infty }& 0& 1& 2& 3\\ \mathrm{\infty }& 3& 0& 1& 2\\ \mathrm{\infty }& 2& 3& 0& 1\\ \mathrm{\infty }& 1& 2& 3& 0\end{array}\right]$

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