A Criterion for the t-Vertex Condition of Graphs

Reichard, Sven

doi:10.1006/jcta.1999.3045

Cited by 13 publications

(13 citation statements)

References 7 publications

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“…Later, Brouwer, Ivanov, and Klin [4] generalized the construction above, obtaining an infinite series of graphs of order 4 k with the 4-vertex condition. These graphs in fact satisfy the 5-vertex condition as was shown by the author in [37].…”

Section: Introductionmentioning

confidence: 68%

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Strongly regular graphs with the $$7$$ 7 -vertex condition

Reichard

2014

J Algebr Comb

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The t-vertex condition, for an integer t ≥ 2, was introduced by Hestenes and 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.

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Section: Introductionmentioning

confidence: 68%

“…The project presented in this paper has quite a long history, stemming from the author's Master's thesis, reflected and extended in [37]. The next steps were done by the author in his Ph.D. thesis [38], fulfilled under the supervision of Mikhail Klin and Felix Lazebnik.…”

Section: A Paper By Christian Pechmentioning

confidence: 99%

Strongly regular graphs with the $$7$$ 7 -vertex condition

Reichard

2014

J Algebr Comb

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“…A first analysis of these graphs was given in [3]. Further steps were taken in [17] and [29]. In the following we collect what is known about the graphs Γ (m) and their subconstituents and what is relevant for this paper:…”

Section: Constructions and Resultsmentioning

confidence: 99%

On a family of highly regular graphs by Brouwer, Ivanov, and Klin

Pech¹,

Pech

2019

Discrete Mathematics

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Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W. Kantor's non-classical GQ(5 2 , 5), are stumbling stones for existing implementations of graph isomorphism tests. They appear to be extremely rare and even once constructed it is difficult to prove their high regularity. Yet some of them, like the McLaughlin graph on 275 vertices and Ivanov's graph on 256 vertices are of profound beauty. This alone makes it an attractive goal to strive for their complete classification or, failing this, at least to get a deep understanding of them. Recently, one of the authors discovered new methods for proving high regularity of graphs. Using these techniques, in this paper we study a classical family of strongly regular graphs, originally discovered by A.E. Brouwer, A.V. Ivanov, and M.H. Klin in the late 80s. We analyze their symmetries and show that they are (3, 5)-regular but not 2-homogeneous. Thus we promote these graphs to the distinguished club of highly regular graphs with few symmetries.We are usually not so much interested into regularities for particular graph types but rather for whole classes.Definition 1.2. A graph is called • ( = m, = n)-regular, if it is T-regular for each graph type T of order (m, n), • ( = m, n)-regular, if it is ( = m, = l)-regular for all m ≤ l ≤ n, • (m, n)-regular, if it is ( = k, n)-regular for all k ≤ m.2010 Mathematics Subject Classification. 05E30(51A50).

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“…One may hope that a deeper utilization of techniques developed by Hestenes and Higman [5], Higman [6], and others (for example, [16]) could provide a new proof that either is computer-free or has a less involved computer search.…”

Section: Discussionmentioning

confidence: 99%

Steiner triple systems satisfying the 4-vertex condition

Kaski

Khatirinejad

Östergård

2011

Des. Codes Cryptogr.

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Higman asked which block graphs of Steiner triple systems of order v satisfy the 4-vertex condition and left the cases v = 9, 13, 25 unsettled. We give a complete answer to this question by showing that the affine plane of order 3 and the binary projective spaces are the only such systems. The major part of the proof is to show that no block graph of a Steiner triple system of order 25 satisfies the 4-vertex condition.

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A Criterion for the t-Vertex Condition of Graphs

Cited by 13 publications

References 7 publications

Strongly regular graphs with the $$7$$ 7 -vertex condition

Strongly regular graphs with the $$7$$ 7 -vertex condition

On a family of highly regular graphs by Brouwer, Ivanov, and Klin

Steiner triple systems satisfying the 4-vertex condition

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