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

The WL-rank of a digraph $\mathrm{\Gamma }$ is defined to be the rank of the coherent configuration of $\mathrm{\Gamma }$. The WL-dimension of $\mathrm{\Gamma }$ is defined to be the smallest positive integer $m$ for which $\mathrm{\Gamma }$ 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 $3$.