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arXiv · 1401.6816

Strongly regular graphs with the 7-vertex condition

Sven Reichard
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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.

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Sven Reichard. 2014-01-27. Strongly regular graphs with the 7-vertex condition. https://arxiv.org/abs/1401.6816

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