The clique graphs of the hexagonal lattice -- an explicit construction and a short proof of divergence

TL;DR

This paper provides an explicit geometric construction of the iterated clique graphs of the hexagonal lattice, demonstrating their divergence and clarifying properties like bounded degrees and clique sizes.

Contribution

It introduces a new explicit geometric method for analyzing clique graphs of the hexagonal lattice, highlighting divergence and clarifying previous observations.

Findings

Clique graphs of Hex diverge under iteration.

Degrees and clique sizes remain bounded despite divergence.

Explicit geometric construction simplifies understanding of clique-divergence.

Abstract

We present a new, explicit and very geometric construction for the iterated clique graphs of the hexagonal lattice $\mathrm{Hex}$ which makes apparent its clique-divergence and sheds light on some previous observations, such as the boundedness of the degrees and clique sizes of $k^n \mathrm{Hex}$ as $n\to\infty$.