On WL-rank and WL-dimension of some Deza dihedrants

TL;DR

This paper investigates the WL-rank and WL-dimension of certain Deza dihedrants, revealing specific rank and dimension properties, and introduces a new family with linearly increasing WL-rank.

Contribution

It establishes the WL-rank and WL-dimension for families of Deza dihedrants and constructs a new infinite family with linearly growing WL-rank.

Findings

Some Deza dihedrants have WL-rank 4 or 5 and WL-dimension 2.

All strictly Deza dihedrants with up to 59 vertices are either circulant or from specific families.

A new infinite family of Deza dihedrants with WL-rank linear in the number of vertices.

Abstract

The WL-rank of a graph $\Gamma$ is defined to be the rank of the coherent configuration of $\Gamma$. The WL-dimension of $\Gamma$ is defined to be the smallest positive integer $m$ for which $\Gamma$ is identified by the $m$-dimensional Weisfeiler-Leman algorithm. We establish that some families of strictly Deza dihedrants have WL-rank $4$ or $5$ and WL-dimension $2$. Computer calculations imply that every strictly Deza dihedrant with at most $59$ vertices is circulant or belongs to one of the above families. We also construct a new infinite family of strictly Deza dihedrants whose WL-rank is a linear function of the number of vertices.