Krein parameters and antipodal tight graphs with diameter 3 and 4

Jurišić, Aleksandar; Koolen, Jack H.

doi:10.1016/s0012-365x(01)00082-6

Cited by 29 publications

(46 citation statements)

References 25 publications

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“…But if the diameter of H is 4, than the best known bound has been weaker: k ≤ r (n − 1)/(2r − 1) . There were no known examples attaining this bound, so there was a conjecture that k ≤ (n − 1)/2 also for this case, see [6, p. 56] or [8]. In this paper we prove that this conjecture is indeed true.…”

Section: Introductionmentioning

confidence: 58%

Valency of Distance-regular Antipodal Graphs with Diameter 4

Miklavič

2002

European Journal of Combinatorics

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Let G be a non-bipartite strongly regular graph on n vertices of valency k. We prove that if G has a distance-regular antipodal cover of diameter 4, then k ≤ 2(n + 1)/5 , unless G is the complement of triangular graph T (7), the folded Johnson graph J (8, 4) or the folded halved 8-cube. However, for these three graphs the bound k ≤ (n − 1)/2 holds. This result implies that only one of a complementary pair of strongly regular graphs can be the antipodal quotient of an antipodal distanceregular graph.

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Section: Introductionmentioning

confidence: 58%

Valency of Distance-regular Antipodal Graphs with Diameter 4

Miklavič

2002

European Journal of Combinatorics

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“…It is well known that a tight graph of diameter 3 is a Taylor graph (see, e.g., [2,Theorem 3.2]). In this case, the neigh borhood of any vertex is a strongly regular graph with k' = 2μ'.…”

Section: Letmentioning

confidence: 99%

On distance-regular graphs in which neighborhoods of vertices are strongly regular

2013

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We consider undirected graphs without loops or multiple edges. Given a vertex a in a graph Γ, let Γ i (a) denote the i neighborhood of a, i.e., the subgraph induced by Γ on the set of all its vertices that are a dis tance of i away from a. Let [a] = Γ 1 (a) and a ⊥ = {a} ∪ [a].A strongly regular graph with parameters (v, k, λ, μ) and with integer eigenvalues k, r, and s, where s < 0, is called a Smith graph if A nonbipartite graph for which the fundamental bound becomes an equality is called tight. The neigh borhood of any vertex in a tight graph is a strongly regular graph with eigenvalues a 1 , bIt is well known that a tight graph of diameter 3 is a Taylor graph (see, e.g., [2, Theorem 3.2]). In this case, the neigh borhood of any vertex is a strongly regular graph with k' = 2μ'.An incidence system with a set of points P and a set of straight lines ᏸ is called an α partial geometry of order (s, t) (denoted by pG α (s, t)) if each line contains s + 1 points; each point lies on t + 1 lines; any two points lie on at most one line; and, for any antiflag (a, l) ∈ (P, ᏸ), there are precisely α lines that pass through a and intersect l. If α = 1, the geometry is called a generalized quadrangle and is denoted by GQ(s, t). The point graph of a geometry is defined on the set of points P, and two points are adjacent if they lie on a line. The point graph of pG α (s, t) is strongly regular with v = (s + 1) , k = s(t + 1), λ = s -1 + t(α -1), and μ = α(t + 1). A strongly regular graph with such parameters for some positive integers α, s, and t is called a pseudogeometric graph for pG α (s, t).Makhnev has proposed a program for the study of distance regular graphs in which the neighborhoods of vertices are strongly regular graphs with given parame ters. This program has been implemented in the case of strongly regular graphs with the eigenvalue 2 (see [3]).An antipodal distance regular graph Γ of diameter 3 has (see [1]) the intersection array {k, μ(r -1), 1; 1, μ, k} and possesses v = r(k + 1) vertices and the spectrum k b

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“…Certain examples of tight graphs are µ-locally the complete multipartite graph K t×n , n, t ∈ N, cf. [10] and [12]. If is 1-homogeneous, then c 2 = µ +1 if and only if is a Terwilliger graph, i.e., is µ-locally K t .…”

Section: Local Regularity Conditionsmentioning

confidence: 99%