Strongly regular graphs with the 7-vertex condition

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The t-vertex condition, for an integer t≥ 2, was introduced by Hestenes and Higman in 1971, providing a combinatorial invariant defined on edges and non-edges of a graph. Finite rank 3 graphs satisfy the condition for all values of t. Moreover, a long-standing conjecture of M. Klin asserts the existence of an integer t₀ such that a graph satisfies the t₀-vertex condition if and only if it is a rank 3 graph. We construct the first infinite family of non-rank 3 strongly regular graphs satisfying the 7-vertex condition. This implies that the Klin parameter t₀ is at least 8. The examples are the point graphs of a certain family of generalised quadrangles.