The automorphism groups and identification of some Generalized Paley Graphs

TL;DR

This work analyzes generalized Paley graphs $X=\mathop{\mathrm{GP}}(q,\frac{q-1}{k})$ with $k\ge 2$, proving that for $q$ sufficiently large relative to $k$ the automorphism group satisfies $\mathrm{Aut}(X)\le \mathrm{AΓL}(1,q)$ and the Weisfeiler–Leman dimension is at most $5$. The approach combines coherent configurations, S-ring theory, and a Babai–Gál–Wigderson-type bound to control neighborhood intersections and vertex distinguishability, enabling an asymptotic identification of these graphs within their class. The paper extends these results to the Van Lint–Schrijver graphs, establishing the same automorphism bound and locating the WL-dimension between $3$ and $5$, with precise determination remaining open in general. Together, these results advance understanding of when automorphism structure and graph identification can be efficiently characterized for Paley-type graphs, with implications for isomorphism testing in this graph family.

Abstract

The family of generalized Paley graphs of prime power order $q$ and degree $(q-1)/k$ is studied. It is shown that the automorphism group of a graph in this family is a subgroup of ${\mathrm{AΓL}}(1,q)$ whenever $q$ is sufficiently large relative to $k$. Furthermore, under the same conditions, the Weisfeiler-Leman dimension of these graphs is proved to be at most $5$. In particular, the same bound holds for the Van Lint-Schrijver graphs.

Theorems & Definitions (21)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 4.1
• Lemma 4.2
• proof