A strongly regular graph with parameters (217,66,15,22)

Abstract

We construct a strongly regular graph with parameters (217,66,15,22), and hence one with the complementary parameters (217,150,105,100). Both parameter sets were marked "?" (existence unknown) in Brouwer's online table on 18 October 2026, with the remark that they are the parameters of the point graph of a hypothetical partial geometry pg(6,10,2). The graph lives on Z31 × {0,…,6}; it is a 7 × 7 block-circulant graph whose connection sets are unions of {0} and cosets of the subgroup of index 6 of F31*. Its strong regularity is proved from the cyclotomic numbers of order 6 over F31. The automorphism group is C2 × (C31:C15), of order 930, with two vertex orbits, of sizes 31 and 186; in particular the graph is not vertex-transitive and not a Cayley graph. The vertex set is partitioned into 31 Delsarte (Hoffman) cliques of size 7, the orbit of size 31 is a Hoffman coclique, and the graph has exactly 31 cliques of size 7; hence it is not geometric, and the existence of pg(6,10,2) remains open. The graph was found by a computer search for graphs with a prescribed intransitive group of automorphisms with few orbits; the cyclotomic description was extracted from it afterwards. The data archive contains the LaTeX source, the graph in graph6 format, a standard-library program that checks the cyclotomic-number proof, a program that rebuilds the graph from its definition and checks it, a standalone SRG verifier, the clean-container run log, the independent verification report, the referee report, and the supplementary report on the cyclotomic family over Z31.