Strongly regular graphs with the 7-vertex condition

TL;DR

This paper constructs an infinite family of non-rank 3 strongly regular graphs satisfying the 7-vertex condition, providing new insights into the Klin conjecture about the characterization of rank 3 graphs.

Contribution

It presents the first known infinite family of such graphs satisfying the 7-vertex condition, challenging previous assumptions about the Klin parameter.

Findings

Constructed the first infinite family of non-rank 3 strongly regular graphs with the 7-vertex condition.

Showed the Klin parameter t_0 is at least 8.

Connected the examples to point graphs of generalized quadrangles.

Abstract

The $t$-vertex condition, for an integer $t\ge 2$, 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 $t_0$ such that a graph satisfies the $t_0$-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 $t_0$ is at least 8. The examples are the point graphs of a certain family of generalised quadrangles.