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A well-known conjecture of Richard Stanley posits that the $h$-vector of theindependence complex of a matroid is a pure ${\mathcal O}$-sequence. Theconjecture has been established for various classes but is open for graphicmatroids. A biconed graph is a graph with two specified `coning vertices', suchthat every vertex of the graph is connected to at least one coning vertex. Theclass of biconed graphs includes coned graphs, Ferrers graphs, and completemultipartite graphs. We study the $h$-vectors of graphic matroids arising frombiconed graphs, providing a combinatorial interpretation of their entries interms of `$2$-weighted forests' of the underlying graph. This generalizesconstructions of Kook and Lee who studied the M\"obius coinvariant (the lastnonzero entry of the $h$-vector) of graphic matroids of complete bipartitegraphs. We show that allowing for partially $2$-weighted forests gives rise toa pure multicomplex whose face count recovers the $h$-vector, establishingStanley's conjecture for this class of matroids. We also discuss how ourconstructions relate to a combinatorial strengthening of Stanley's Conjecture(due to Klee and Samper) for this class of matroids.

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