Triangle-Orbit Congruences for a Putative Conway 99-Graph: Order-Three and Involution Automorphisms

Abstract:We study necessary conditions on a putative strongly regular graph with parameters (99, 14, 1, 2) — a Conway 99-graph, the smallest undecided case of the form (v, k, 1, 2). We prove an elementary congruence: for an automorphism of prime order p of a finite graph, the number of triangles is congruent modulo p to the number of triangles preserved setwise. Combining this with a spectral projector for the 3-eigenspace, we obtain necessary conditions in the surviving automorphism cases: (i) a fixed-point-free order-3 automorphism has triangle-orbit parameter T ∈ {6, 27}; (ii) an order-3 automorphism fixing a triangle has t ≡ 2 (mod 21) and, since every triangle-orbit is a free orbit, t = 2; (iii) an involution fixing exactly one vertex has e ≡ 0 (mod 7) with e odd, so e ∈ {7, 21, 35, 49}, conditional on a cited fixed-point classification. We also prove the first of two quotient identities, Σ P