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

When  = number of vertices,  = number of edges, and  = 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  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](...).