The rank polynomial of a general graph is the function defined by
where the sum is taken over all subgraphs (i.e., edge sets) and the rank and co-rank of the subgraph is given by
for a subgraph with vertices, edges, and connected components (Biggs 1993, p. 73).
The rank polynomial is multiplicative over graph components, so for a graph having connected components , , ..., the rank polynomial of itself is given by
It is given in terms of the Tutte polynomial as
The chromatic polynomial and rank polynomial of a general graph with vertices are related by
(Biggs 1993, p. 75).
where is the number of edges of the graph (Biggs 1993, p. 74).
The following table summarizes rank polynomials for some common classes of graphs.
|ladder rung graph|
The following table summarizes recurrence equations for rank polynomials of common classes of graphs.
Nonisomorphic graphs do not necessarily have distinct rank polynomials. The following table summarizes some co-rank graphs.
|, , , , , , path graph|
|triangle graph, , , , ,|
|, , , , , , , , -fiveleaper graph, -fiveleaper graph, -knight graph, 2-ladder rung graph, path graph|
|claw graph, , , , , , , , , , , , , , , -fiveleaper graph, 3-ladder rung graph, path graph|
|paw graph, , , , , , , , )|
|square graph, , , , ,|
|diamond graph, , , ,|
|tetrahedral graph, , , ,|