Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs

Abstract

Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalent to the Ehrhart polynomial of Q, which in turn is equivalent to the h-vector of any triangulation of Q. It follows that the interior polynomials of (V,E) and its transpose (E,V) agree. When G is a complete bipartite graph, our result recovers a well known hypergeometric identity due to Saalschütz. It also implies that certain extremal coefficients in the Homfly polynomial of a special alternating link can be read off of an associated Floer homology group.

30 pages, 5 figures

Cited by in corpus (7)

• PQ-type adjacency polytopes of join graphs
• Root polytopes and Jaeger-type dissections for directed graphs
• Reflexive polytopes arising from bipartite graphs with $\gamma$-positivity associated to interior polynomials
• Edge rings of bipartite graphs with linear resolutions
• A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids
• $h^{\ast }$-vectors of graph polytopes using activities of dissecting spanning trees
• Tight contact structures on Seifert surface complements