On highly regular strongly regular graphs

Pech, Christian

doi:10.5802/alco.183

Pech, Christian ¹

¹ Radebeul, Germany

Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 843-878.

Abstract

In this paper we unify several existing regularity conditions for graphs, including strong regularity, $k$ -isoregularity, and the $t$ -vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the $7$ -vertex condition fulfills an even stronger condition, $(3, 7)$ -regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank $3$ strongly regular graphs satisfying the $6$ -vertex condition. This strengthens and generalizes previous results by Reichard.

Received: 2017-12-19
Revised: 2021-02-28
Accepted: 2021-05-06
Published online: 2021-11-03

DOI: 10.5802/alco.183

Classification: 05E30, 51E12
Keywords: Strongly regular graphs, invariants, $k$-isoregularity, $t$-vertex condition, partial quadrangles, generalized quadrangles, partial linear spaces

Author's affiliations:

Pech, Christian ¹

¹ Radebeul, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Pech, Christian. On highly regular strongly regular graphs. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 843-878. doi : 10.5802/alco.183. https://alco.centre-mersenne.org/articles/10.5802/alco.183/

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