GraphRank returns rank of a graph G which is the number of vertices minus the number of connected components of G. If a set of edges (arcs) E of the (di)graph is specified, then the rank is number of vertices minus the number of connected components of the spanning subgraph of G with edge set E.

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

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

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

$3$

$\mathrm{GraphRank}\left(G\,\left\{\left\{1\,2\right\}\,\left\{3\,4\right\}\right\}\right)$

$2$