Biconed graphs, weighted forests, and h-vectors of matroid complexes

TL;DR

This paper proves Stanley's conjecture for the $h$-vectors of graphic matroids derived from biconed graphs by interpreting entries as counts of weighted forests, thus establishing a combinatorial structure for these matroids.

Contribution

It introduces a combinatorial interpretation of $h$-vector entries for biconed graph matroids using weighted forests, confirming Stanley's conjecture for this class.

Findings

Provides a combinatorial interpretation of $h$-vectors in terms of weighted forests.

Establishes Stanley's conjecture for graphic matroids of biconed graphs.

Connects constructions to a strengthened version of Stanley's conjecture.

Abstract

A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${\mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified `coning vertices', such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of `$2$-weighted forests' of the underlying graph. This generalizes constructions of Kook and Lee who studied the M\"obius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially $2$-weighted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanley's conjecture for this class of matroids. We also discuss how our constructions relate to a combinatorial strengthening of Stanley's Conjecture (due to Klee and Samper) for this class of matroids.