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.

These polynomials were discovered by Béla Bollobás and Oliver Riordan (2001, 2002).^[1][2]

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 ${\displaystyle G}$.

• Graph invariant