Chromatic symmetric functions of conjoined graphs
E.Y.J. Qi, D.Q.B. Tang, D.G.L. Wang
TL;DR
This paper studies the chromatic symmetric functions of new graph classes formed by conjoining rooted graphs with paths, providing positive expansions and proposing a conjecture on e-positivity.
Contribution
It introduces path-conjoined, spider-conjoined, and chain-conjoined graphs, and derives positive e-expansions for their chromatic symmetric functions using a novel composition method.
Findings
Positive e-expansions for clique-path-cycle graphs
Positive e-expansions for path-clique-path graphs
Positive e-expansions for clique-clique-path graphs
Abstract
We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using the composition method developed by Zhou and the third author recently, we obtain neat positive -expansions for the chromatic symmetric functions of clique-path-cycle graphs, path-clique-path graphs, and clique-clique-path graphs. We pose the -positivity conjecture for hat-chains.
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Taxonomy
TopicsGraph theory and applications · Graph Labeling and Dimension Problems · Advanced Graph Theory Research