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

Reichard, Sven

doi:10.1007/s10801-014-0554-1

Cited by 7 publications

(8 citation statements)

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“…It is well known that a strongly regular graph Γ is (3,4)-regular if and only if its subconstituents Γ i (v) are strongly regular with parameters independent from v ∈ V (Γ) (for a proof see, e.g., [31,Proposition 4]). Thus, Ivanov's graph Γ (4) is (3, 4)-regular.…”

Section: Constructions and Resultsmentioning

confidence: 99%

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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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Section: Constructions and Resultsmentioning

confidence: 99%

“…2005: Reichard shows that the point graphs of GQ(q, q 2 ) are (2, 7)-regular (cf. [31]). 2007: CP shows that the point graphs of PQ(q − 1, q 2 , q 2 − q) are (2, 6)-regular.…”

Section: Introductionmentioning

confidence: 99%

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

Pech¹,

Pech

2019

Discrete Mathematics

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“…Reichard [31] showed that the collinearity graphs of generalized quadrangles satisfy the 5-vertex condition, and that the collinearity graphs of generalized quadrangles GQ(s, s 2 ) satisfy the 7-vertex condition.…”

Section: Generalized Quadranglesmentioning

confidence: 99%

Strongly regular graphs satisfying the 4-vertex condition

Brouwer¹,

Ihringer²,

Kantor³

2021

Preprint

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“…Following Hestenes and Higman [14], a strongly regular graph Γ is said to enjoy the t-vertex condition if, for any graph G on at most t nodes and any two distinguished nodes a, b ∈ V (G) the number of graph homomorphisms from G to Γ mapping a to x and b to y depends only on whether x and y are equal, adjacent, or non-adjacent. A recent investigation on this topic is Reichard's paper [24].…”

Section: The Vector Space Of Scaffolds Of Order Twomentioning

confidence: 99%

Scaffolds: a graph-based system for computations in Bose-Mesner algebras

Martin

2020

Preprint

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Let X be a finite set and let Mat X (C) denote the algebra of matrices with rows and columns indexed by X and entries from the complex numbers acting on C X with standard basis {x | x ∈ X}. For a digraph G = (V (G), E(G)), function R : [m] → V (G) with r j := R(j), and a function w from the arcs of G to Mat X (C), we define the "scaffold" S(G, R; w) as the sum over all functions ϕ from V (G) to X of the m-fold tensors ϕ(r 1 )⊗ ϕ(r 2 )⊗• • •⊗ ϕ(r m ) scaled by the product of the entries w(e) ϕ(a),ϕ(b) over all arcs e = (a, b) of G. If Γ is a digraph with adjacency matrix A, m = 0 and w(e) = A for all e, this simply counts all digraph homomorphisms from G to Γ. If instead, Γ is a distance-regular graph with i th distance matrix A i and G is a star graph K 1,m with central node a and degree one nodes r 1 , . . . , r m , the scaffold S(G, R; w) encodes generalized intersection numbersScaffolds also arise in the the theory of link invariants and spin models: the partition function of a link diagram is encoded as a scaffold of order zero and invariance under the three Reidemeister moves are encoded as identities among scaffolds of order two and three. These diagrams were introduced in the late 1980s by Arnold Neumaier to aid in computations in the theory of distance-regular graphs. Various authors have used scaffolds to systematize complex calculations of parameters in association schemes. Prioritizing the tutorial value of the paper, we briefly revisit results of Dickie, Suzuki, and Terwilliger using this diagrammatic formalism. Commonly employed transformations are presented as a basic system of "moves" on these diagrams that preserve their value and certain natural actions of the Bose-Mesner algebra on nodes and edges of a scaffold given by Terwilliger and Jaeger are reviewed. Many of the results presented here are not new; our goal is to collect and present, in a uniform fashion, Neumaier's original idea extended to tensors and its use by various authors. Sometimes the term "star-triangle diagram" appears for what we, in this paper, call "scaffolds".When one fixes the diagram G and root nodes R but allows the edge weights to vary over matrices in a given coherent algebra A, the vector space W((G, R); A) spanned by all resulting tensors seems worthy of study. We show that W((H, R ′ ); A) is contained in W((G, R); A) when (H, R ′ ) is a rooted minor of (G, R) and examine several important spaces of this form with three root nodes in connection with the Terwilliger algebra.It is not surprising that some scaffold identities are more intuitive than others: for instance, much more is known about distance-regular graphs than is known about Q-polynomial ("cometric") association schemes. Connecting duality in the theory of association schemes to duality of circular planar graphs, we present a conjecture dealing with dual pairs of planar scaffolds which points to a tool for the generation of new identities.

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

Cited by 7 publications

References 28 publications

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

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

Strongly regular graphs satisfying the 4-vertex condition

Scaffolds: a graph-based system for computations in Bose-Mesner algebras

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