Geodesic
Strongly regular graphs with the $7$-vertex condition
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 817-842.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The $t$-vertex condition, for an integer $t\geq 2$, was introduced by M. D. Hestenes and D. G. Higman [SIAM Am. Math. Soc. Proc. 4, 41--160 (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 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 present 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 generalized quadrangles.
Classification : 05E30, 05C85
Keywords: strongly regular graph, $t$-vertex condition, generalized quadrangle 
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     title = {Strongly regular graphs with the $7$-vertex condition},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2015__41_3_a4/}
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Reichard, Sven. Strongly regular graphs with the $7$-vertex condition. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 817-842. http://geodesic.mathdoc.fr/item/JAC_2015__41_3_a4/