Delsarte clique graphs

Bang, Sejeong; Hiraki, Akira; Koolen, Jack H.

doi:10.1016/j.ejc.2005.04.015

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Introduction2

Some Useful Results1

Properties Of Geometric Distance-regular Graphs1

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Cited by 27 publications

(31 citation statements)

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“…A special case of the diameter bound given by A.A. Ivanov ([17], cf. [7, Theorem 5.9.8]) gives that the diameter of a distance-regular graph with valency k and c 2 2 satisfies D 4 k . As there are only finitely many connected nonisomorphic k-regular graphs with diameter at most 4 k , we obtain the following theorem.…”

Section: Some Useful Resultsmentioning

confidence: 95%

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On distance-regular graphs with smallest eigenvalue at least −m

Koolen

Bang

2010

Journal of Combinatorial Theory, Series B

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Section: Some Useful Resultsmentioning

confidence: 95%

“…First, we recall the notion of a Delsarte pair as introduced by S. Bang, A. Hiraki and J. Koolen [2].…”

Section: Properties Of Geometric Distance-regular Graphsmentioning

confidence: 99%

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On distance-regular graphs with smallest eigenvalue at least −m

Koolen

Bang

2010

Journal of Combinatorial Theory, Series B

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“…1 It will be shown that each member of C is a maximal clique if d(Γ ) > 1. See a remark preceding Lemma 5.…”

Section: Introductionmentioning

confidence: 99%

“…When C consists of Delsarte cliques, it is called a Delsarte clique graph with parameters (s, c) in [1,2], and Delsarte clique graphs with parameters (s, 1) are called geometric in [3]. For examples of completely regular clique graphs, see [1] and Sect.…”

Section: Introductionmentioning

confidence: 99%

Completely regular clique graphs, II

Suzuki

2015

J Algebr Comb

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Let Γ = (X, R) be a connected graph. Then Γ is said to be a completely regular clique graph of parameters (s, c) with s ≥ 1 and c ≥ 1, if there is a collection C of completely regular cliques of size s + 1 such that every edge is contained in exactly c members of C. In the previous paper (Suzuki in J Algebr Combin 40:233-244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to C is distance-regular if and only if every T (C)-module of endpoint zero is thin for all C ∈ C. We also discuss the relation between a T (C)-module of endpoint 0 and a T (x)-module of endpoint 1 and study examples of completely regular clique graphs.

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