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October 12, 2016October 12, 2016 thcscience_admin

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[edit]

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

Formal definition[edit]

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[edit]

Graph invariant

References[edit]

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

source: https://en.wikipedia.org/wiki/Bollob%C3%A1s%E2%80%93Riordan_polynomial

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