The e positivity of some new classes of graphs

TL;DR

This paper introduces two new classes of graphs, suns and dumbbells, and investigates their chromatic symmetric functions, establishing their e-positivity and connections with other graph classes.

Contribution

The paper defines suns and dumbbells, analyzes their chromatic symmetric functions, and demonstrates their e-positivity, expanding understanding of graph classes and their properties.

Findings

Suns and dumbbells are e-positive.

Connections established with other graph classes.

Enhanced understanding of chromatic symmetric functions.

Abstract

We introduce two classes of graphs - suns and dumbbells, both with few variations and explore their chromatic symmetric function and its $e$-positivity. We also give many connections of these two classes with other classes of connected graphs.