Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes   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 `edge-rooted 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 edge-rooted forests gives rise to a
pure multicomplex whose face count recovers the $h$-vector, establishing
Stanley's conjecture for this class of matroids.

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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-bio
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The categories of this paper are: physics
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The categories of this paper are: q-alg