Strongly Regular Graphs of Rank Four
William H. Allen
TL;DR
This work classifies strongly regular graphs that admit a non-affine rank-four group of automorphisms by leveraging a complete classification of non-affine rank-four primitive groups and testing their orbital graphs. The approach combines group-theoretic reduction (via Cuypers’ and related classifications) with computational verification (GAP/GRAPE) of distance-regularity and strong regularity of orbital graphs, yielding an explicit list of SRGs in Table 1, including families from alternating-socle, parabolic, and nonsingular subspace actions, as well as notable examples like the generalized Johnson graphs and polar-graph-derived graphs. The main findings identify several exceptional SRGs with precise parameters, such as $(784,243,82,72)$, $(35,18,9,9)$, $(120,63,30,36)$, and families like $NU_n(3)$, $NU_n(4)$, and $NO_{2m+1}^{\pm}(5)$, each associated with specific almost-simple group actions. These results deepen the understanding of how large automorphism groups constrain the structure of SRGs and highlight deep connections between finite groups of Lie type, incidence geometries, and algebraic-combinatorial graphs, with explicit parameter sets and constructions documented for further exploration.
Abstract
Strongly regular graphs are regular graphs with a constant number of common neighbours between adjacent vertices, and a constant number of common neighbours between non-adjacent vertices. These graphs have been of great interest over the last few decades and often give rise to interesting groups of automorphisms. In this paper we take a reverse approach, and leverage strong classification results on rank four permutation groups to classify the strongly regular graphs which yield such groups as a group of automorphisms.