Distance-regular graphs

van Dam, Edwin R.; Koolen, Jack H.; Tanaka, Hajime

doi:10.48550/arxiv.1410.6294

Cited by 19 publications

(34 citation statements)

References 528 publications

(874 reference statements)

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“…). In view of Lemma 2.1, we consider the following subset of R: (7) π(Γ) = {t ∈ R : K t is positive semidefinite}.…”

Section: Algebraic Probability Spaces Distance-regular Graphs and The...mentioning

confidence: 99%

“…[36]. Another important class of graphs to consider here is that of distance-regular graphs [3,4,5,7], which generalize distancetransitive (i.e., two-point homogeneous) graphs. Among many other applications and links, these graphs have been often used as test instances for problems related to random walks on general graphs; see [7] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Another important class of graphs to consider here is that of distance-regular graphs [3,4,5,7], which generalize distancetransitive (i.e., two-point homogeneous) graphs. Among many other applications and links, these graphs have been often used as test instances for problems related to random walks on general graphs; see [7] and the references therein. Hora [13] proved CLTs for several families of distance-regular graphs, including the Hamming graphs (which are also Cayley graphs) and the Johnson graphs, and obtained various distributions, such as the Gaussian, Poisson, geometric, and the exponential distributions.…”

Section: Introductionmentioning

confidence: 99%

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Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Koohestani,

Obata,

Tanaka

2021

Preprint

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We determine the possible scaling limits in the quantum central limit theorem with respect to the Gibbs state, for a growing distance-regular graph that has so-called classical parameters with base unequal to one. We also describe explicitly the corresponding weak limits of the normalized spectral distribution of the adjacency matrix. We demonstrate our results with the known infinite families of distance-regular graphs having classical parameters and with unbounded diameter.

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“…). In view of Lemma 2.1, we consider the following subset of R: (7) π(Γ) = {t ∈ R : K t is positive semidefinite}.…”

Section: Algebraic Probability Spaces Distance-regular Graphs and The...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Koohestani,

Obata,

Tanaka

2021

Preprint

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“…See, e.g., [10,12,15,27] for more information on distance-regular graphs. We say that (X, R) is Q-polynomial (or cometric) with respect to the ordering…”

Section: R∈r0mentioning

confidence: 99%

Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials

Bannai,

Tanaka

et al. 2020

Preprint

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Relative t-designs in the n-dimensional hypercube Qn are equivalent to weighted regular t-wise balanced designs, which generalize combinatorial t-(n, k, λ) designs by allowing multiple block sizes as well as weights. Partly motivated by the recent study on tight Euclidean t-designs on two concentric spheres, in this paper we discuss tight relative t-designs in Qn supported on two shells. We show under a mild condition that such a relative t-design induces the structure of a coherent configuration with two fibers. Moreover, from this structure we deduce that a polynomial from the family of the Hahn hypergeometric orthogonal polynomials must have only integral simple zeros. The Terwilliger algebra is the main tool to establish these results. By explicitly evaluating the behavior of the zeros of the Hahn polynomials when they degenerate to the Hermite polynomials under an appropriate limit process, we prove a theorem which gives a partial evidence that the non-trivial tight relative t-designs in Qn supported on two shells are rare for large t.

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“…As κ ′ (G) > κ(G) for super-connected graphs, it is natural to ask what the super-connectivity of the Johnson graph J(n, k) is. In [27], van Dam et al also asked about the minimum number of vertices that need to be deleted to disconnect a distance-regular graph with diameter at least three such that each resulting component has at least two vertices (Problem 41). In Theorem 10, we verify their claimed value in the case of Johnson graphs.…”

Section: Introductionmentioning

confidence: 99%

The super-connectivity of Johnson graphs

Ekinci,

Gauci

2019

Preprint

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For positive integers n, k and t, the uniform subset graph G(n, k, t) has all k-subsets of {1, 2, . . . , n} as vertices and two k-subsets are joined by an edge if they intersect at exactly t elements. The Johnson graph J(n, k) corresponds to G(n, k, k − 1), that is, two vertices of J(n, k) are adjacent if the intersection of the corresponding k-subsets has size k − 1. A super vertex-cut of a connected graph is a set of vertices whose removal disconnects the graph without isolating a vertex and the superconnectivity is the size of a minimum super vertex-cut. In this work, we fully determine the super-connectivity of the family of Johnson graphs J(n, k) for n ≥ k ≥ 1.

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Distance-regular graphs

Abstract: This is a survey of distance-regular graphs. We present an introduction to distanceregular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN ' [Brouwer, A.E.

Cited by 19 publications

References 528 publications

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials

The super-connectivity of Johnson graphs

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