Important: The networks package has been deprecated.Use the superseding command GraphTheory[RankPolynomial] instead.

When $n\left(G\right)$ = number of vertices, $m\left(G\right)$ = number of edges, and $c\left(G\right)$ = number of components, one defines rank(G) = n(G) - c(G) and corank(G) = m(G) - rank(G).

The rank polynomial is a sum over all subgraphs H of G of x^(rank(G) - rank(H)) y^corank(H).

The coefficient of $x^i{y}^j$ in the rank polynomial is thus the number of spanning subgraphs of G having i more components than G and having a cycle space of dimension j.

This routine is normally loaded via the command with(networks) but may also be referenced using the full name networks[rankpoly](...).

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

$G≔\mathrm{complete}\left(4\right)\:$

$\mathrm{rankpoly}\left(G\,x\,y\right)$

$x^3+y^3+6x^2+4xy+6y^2+15x+15y+16$

$\mathrm{rankpoly}\left(G\,1\,1\right)$

$64$

$\mathrm{rankpoly}\left(G\,1\,0\right)$

$38$

$\mathrm{rankpoly}\left(G\,0\,1\right)$

$38$

$\mathrm{rankpoly}\left(G\,0\,0\right)$

$16$