Clocks are $e$-positive

TL;DR

This paper proves the $e$-positivity of all clock graphs $ heta_{ab2}$, advancing the understanding of graph positivity properties through composition methods and transformation analysis.

Contribution

It establishes the $e$-positivity of clock graphs $ heta_{ab2}$, extending previous results on cycle-chord and theta graphs using novel composition techniques.

Findings

Proves $e$-positivity of clock graphs $ heta_{ab2}$

Introduces a composition-based method for analyzing graph positivity

Analyzes fibers of partial reversal transformations on compositions

Abstract

Along with his confirmation of the $e$-positivity of all cycle-chord graphs $\theta_{ab1}$, the third author conjectured the $e$-positivity of all theta graphs $\theta_{abc}$. In this paper, we establish the $e$-positivity of all clock graphs $\theta_{ab2}$ by using the composition method. The key idea is to investigate the fibers of certain partial reversal transformation on compositions with all parts at least $2$.