On coefficients of the interior and exterior polynomials

TL;DR

This paper investigates the coefficients of interior and exterior polynomials associated with bipartite graphs, establishing new properties like interpolation, monicity, and linear combination representations, and connecting these polynomials to graph invariants and knot theory.

Contribution

It proves that interior polynomials are interpolating, extends degree bounds for graphs with 2-vertex cuts, and shows interior polynomials form a basis for certain bipartite graph families.

Findings

Interior polynomials are interpolating.

Interior polynomials for balanced bipartite graphs are monic.

Exterior polynomials are also interpolating.

Abstract

The interior polynomial and the exterior polynomial are generalizations of valuations on $(1/\xi,1)$ and $(1,1/\eta)$ of the Tutte polynomial $T_G(x,y)$ of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let $G=(V\cup E, \varepsilon)$ be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in $V$ or $E$. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a `dual' result on the coefficient of the linear term of the exterior polynomial: if $G-e$ is connected for each $e\in E$, then the coefficient of the linear term of the exterior polynomial is $|V|-1$. Interior and exterior polynomials for some families of bipartite graphs are computed.