{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T15:21:36Z","timestamp":1767453696457,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of $e$-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph $G$ centred at a vertex $v$. We then apply our methods to $y_{G : v}$ and find new families of unit interval graphs that are $(e)$-positive, a stronger condition than classical $e$-positivity, thus confirming new cases of the $(3+1)$-free conjecture of Stanley and Stembridge.\r\nIn our study of $y_{G : v}$, we also describe methods of constructing new $e$-positive graphs from given $(e)$-positive graphs and classify the $(e)$-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.<\/jats:p>","DOI":"10.37236\/12319","type":"journal-article","created":{"date-parts":[[2024,10,31]],"date-time":"2024-10-31T17:02:16Z","timestamp":1730394136000},"source":"Crossref","is-referenced-by-count":2,"title":["The Chromatic Symmetric Function of a Graph Centred at a Vertex"],"prefix":"10.37236","volume":"31","author":[{"given":"Farid","family":"Aliniaeifard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Victor","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stephanie","family":"Van Willigenburg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,10,18]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p22\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i4p22\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,31]],"date-time":"2024-10-31T17:02:17Z","timestamp":1730394137000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i4p22"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,18]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,10,3]]}},"URL":"https:\/\/doi.org\/10.37236\/12319","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,10,18]]},"article-number":"P4.22"}}