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 ${\displaystyle G}$ is given by

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

where the sum runs over all the spanning subgraphs ${\displaystyle F}$ and

• ${\displaystyle v(G)}$ is the number of vertices of ${\displaystyle G}$;
• ${\displaystyle e(G)}$ is the number of its edges of ${\displaystyle G}$;
• ${\displaystyle k(G)}$ is the number of components of ${\displaystyle G}$;
• ${\displaystyle r(G)}$ is the rank of ${\displaystyle G}$, such that ${\displaystyle r(G)=v(G)-k(G)}$;
• ${\displaystyle n(G)}$ is the nullity of ${\displaystyle G}$, such that ${\displaystyle n(G)=e(G)-r(G)}$;
• ${\displaystyle bc(G)}$ is the number of connected components of the boundary of ${\displaystyle G}$.

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