3 research outputs found
A Linear Algebraic Method on the Chromatic Symmetric Function
The Stanley-Stembridge conjecture is a longstanding conjecture that has evaded proof for nearly 30 years. Concerned with the e-basis expansions of the chromatic symmetric functions of unit-interval graphs, this conjecture has served as a significant motivator of research in algebraic graph theory in recent years. We summarize a great deal of the existing work done in favor of this conjecture, giving an overview of the various techniques that have previously been used in the study of this problem. Moreover, we develop a novel technique using methods from linear algebra and use it to obtain an e-basis expansion of graphs known as single clique-blowups of paths. Using this same method, we use this result to prove the e-positivity of double clique-blowups of paths, a previously unknown result
Biconed graphs, edge-rooted forests, and h-vectors of matroid complexes
A well-known conjecture of Richard Stanley posits that the -vector of the
independence complex of a matroid is a pure -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 -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 -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 -vector, establishing
Stanley's conjecture for this class of matroids.Comment: 15 pages, 3 figures; V2: added omitted author to metadat