On the imaginary parts of chromatic root

TL;DR

This paper investigates the imaginary parts of chromatic roots of graphs, proving they can grow linearly with the graph size and showing that most random graphs have non-real chromatic roots.

Contribution

It establishes the linear growth of the maximum imaginary part of chromatic roots and demonstrates that almost all Erdős-Rényi random graphs have non-real roots.

Findings

Maximum imaginary part can grow linearly with graph order

Almost all Erdős-Rényi graphs have non-real chromatic roots

Provides new insights into the distribution of chromatic roots

Abstract

While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (that is, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We also show that for any fixed $p \in (0,1)$, almost every random graph $G$ in the Erd\"os-R\'enyi model has a non-real root.