The rank polynomial
of a general graph
is the function defined by

(1)

where the sum is taken over all subgraphs (i.e., edge sets) and the rank and co-rank of the subgraph is given by

(2)

(3)

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

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, pp. 73,
97, and 101, 1993.Godsil, C. and Royle, G. "Rank Polynomial"
and "Evaluations of the Rank Polynomial." §15.9-15.10 in Algebraic
Graph Theory. New York: Springer-Verlag, pp. 355-358, 2001.