On WL-rank and WL-dimension of some Deza circulant graphs

TL;DR

This paper classifies Deza circulant graphs with WL-rank 4, shows they have WL-dimension at most 3, and explores families with higher WL-rank but bounded WL-dimension, advancing understanding of their symmetry properties.

Contribution

It provides a classification of Deza circulant graphs with WL-rank 4 and bounds their WL-dimension, also analyzing families with higher WL-rank.

Findings

Deza circulant graphs of WL-rank 4 are classified.

All such graphs have WL-dimension at most 3.

Some families have WL-rank 5 or 6 with WL-dimension at most 3.

Abstract

The WL-rank of a digraph $\Gamma$ 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 classify the Deza circulant graphs of WL-rank $4$. In additional, it is proved that each of these graphs has WL-dimension at most $3$. Finally, we establish that some families of Deza circulant graphs have WL-rank $5$ or $6$ and WL-dimension at most $3$.