Bollobás–Riordan polynomial

The Bollobás–Riordan polynomial can mean a 3-variable polynomial invariant of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing the Tutte polynomial. These polynomials were discovered by Bollobás and Riordan (2001, 2002).

The 3-variable Bollobás–Riordan polynomial is given by

$RG(x,y,z) =\sum_F x^{r(G)-r(F)}y^{n(F)}z^{k(F)-bc(F)%2Bn(F)}$

where

v(G) is the number of vertices of G;
e(G) is the number of its edges of G;
k(G) is the number of components of G;
r(G) is the rank of G such that r(G) = v(G) − k(G);
n(G) is the nullity of such that n(G) = e(G) − r(G);
bc(G) is the number of connected components of the boundary of G;

References

Bollobás, Béla; Riordan, Oliver (2001), "A polynomial invariant of graphs on orientable surfaces", Proceedings of the London Mathematical Society. Third Series 83 (3): 513–531, doi:10.1112/plms/83.3.513, ISSN 0024-6115, MR 1851080
Bollobás, Béla; Riordan, Oliver (2002), "A polynomial of graphs on surfaces", Mathematische Annalen 323 (1): 81–96, doi:10.1007/s002080100297, ISSN 0025-5831, MR 1906909