Bollobás–Riordan polynomial

The Bollobás–Riordan polynomial can mean a 3-variable invariant polynomial of graphs on orientable surfaces, or a more general 4-variable invariant of ribbon graphs, generalizing the Tutte polynomial.

History

These polynomials were discovered by Béla Bollobás and Oliver Riordan (2001, 2002).

Formal definition

The 3-variable Bollobás–Riordan polynomial of a graph $G$ is given by

$R_{G}(x,y,z)=\sum _{F}x^{r(G)-r(F)}y^{n(F)}z^{k(F)-bc(F)+n(F)}$,

where the sum runs over all the spanning subgraphs $F$ and

• $v(G)$ 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 $G$, such that $n(G)=e(G)-r(G)$;
• $bc(G)$ is the number of connected components of the boundary of $G$.

See also

• Graph invariant

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