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networks

 rankpoly
 (Whitney) rank polynomial of an undirected graph

 Calling Sequence rankpoly(G, x, y)

Parameters

 G - undirected graph or network x - rank variable in rank poly y - corank variable in rank poly

Description

 • 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](...).

Examples

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

 > $\mathrm{with}\left(\mathrm{networks}\right):$
 > $G≔\mathrm{complete}\left(4\right):$
 > $\mathrm{rankpoly}\left(G,x,y\right)$
 ${{x}}^{{3}}{+}{{y}}^{{3}}{+}{6}{}{{x}}^{{2}}{+}{4}{}{x}{}{y}{+}{6}{}{{y}}^{{2}}{+}{15}{}{x}{+}{15}{}{y}{+}{16}$ (1)
 > $\mathrm{rankpoly}\left(G,1,1\right)$
 ${64}$ (2)
 > $\mathrm{rankpoly}\left(G,1,0\right)$
 ${38}$ (3)
 > $\mathrm{rankpoly}\left(G,0,1\right)$
 ${38}$ (4)
 > $\mathrm{rankpoly}\left(G,0,0\right)$
 ${16}$ (5)