On locally $GQ(s,t)$ graphs with strongly regular $\mu $-subgraphs

Abstract

The connected locally GQ(s, t) graphs are studied in which every µsubgraph is a known strongly regular graph (i.e., K m,m for a positive integer m, the Moore graph with parameters (k 2 + 1, k, 0, 1), k = 2, 3, or 7, the Clebsch graph, the Gewirtz graph, the Higman-Sims graph, or the second neighborhood (with parameters (77, 16, 0, 4)) of a vertex in the Higman-Sims graph). It is proved that if Γ is a strongly regular locally GQ(s, t) graph in which every µ-subgraph is isomorphic to a known strongly regular graph ∆, then one of the following statements is true: (1) ∆ = K t+1,t+1 and either s = 1 and Γ = K 3×(t+1) , or s = 4, t = 1, and Γ is a quotient of the Johnson graph J(10, 5), or s = t = 1, 2, 3, 8, 13; (2) ∆ is a Petersen graph and Γ is a unique locally GQ(2, 2) graph with parameters (28, 15, 6, 10); (3) ∆ is the Gewirtz graph and Γ is the McLaughlin graph.