Antipodal covering graphs

Gardiner, A.

doi:10.1016/0095-8956(74)90072-0

Cited by 73 publications

(44 citation statements)

References 7 publications

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“…By Gardiner [7], in an antipodal distance-regular graph with diameter D a vertex x, which is at distance i ≤ D/2 from one vertex in an antipodal class, is at distance D − i from all other vertices in this antipodal class, hence…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

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Jurišić

Koolen

2003

Journal of Algebraic Combinatorics

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Section: Proof Of the Main Resultsmentioning

confidence: 99%

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Jurišić

Koolen

2003

Journal of Algebraic Combinatorics

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“…By analogy, each antipodal (n − 1)-fold cover of the complete graph K n gives rise to a Moore graph of valency n and diameter two [12] and any antipodal k-fold cover of a complete bipartite graph K k,k yields an affine plane. We know of no other antipodal distance-regular graphs with r = k. Thus it is natural to expect stronger bounds on w unless the parameters are particularly nice.…”

Section: The Q-antipodal Case and Dismantlabilitymentioning

confidence: 99%

Imprimitive cometric association schemes: Constructions and analysis

2007

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Dualizing the "extended bipartite double" construction for distance-regular graphs, we construct a new family of cometric (or Q-polynomial) association schemes with four associate classes based on linked systems of symmetric designs. The analysis of these new schemes naturally leads to structural questions concerning imprimitive cometric association schemes, some of which we answer with others being left as open problems. In particular, we prove that any Q-antipodal association scheme is dismantlable: the configuration induced on any subset of the equivalence classes in the Q-antipodal imprimitivity system is again a cometric association scheme. Further examples are explored.

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“…We continue this latter tradition and prove the following theorem-which was recently proved independently by Ivanov et al using a computer [10]. Theorem 6 , P 3 (4), P 4 (4).…”

Section: Introductionmentioning

confidence: 99%

“…K 6 is the complete graph, and K s 5 the complete bipartite graph, of valency five. Q 5 is the 5-dimensional cube, and Ds is the antipodal quotient of Q 5 .…”

Section: Introductionmentioning

confidence: 99%

“…f/ u is the incidence graph of points and blocks in the unique symmetric 2-(l 1,5,2) design. Z 6 has as vertices the thirty-six subgroups of order 20 in S 6 , two such being adjacent when they intersect in a subgroup of order 4 ([1, p. 153]). P 3 (4) and P 4 (4) are the incidence graphs respectively of the projective plane of order 4 and the classical generalised quadrangle of order (4,4) associated with PSp (4,4).…”

Section: Introductionmentioning

confidence: 99%

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Distance-transitive graphs of valency five

Gardiner

Praeger

1987

Proceedings of the Edinburgh Mathematical Society

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If u and v are vertices of the (finite, connected) graph Γ, let d(u, v) denote the length of the shortest path joining u to v in Γ. The graph Γ is said to be distance-transitive if whenever d(u, v) = d(u′, v′), there exists an automorphism g of Γ such that ug = u′ and if vg = v′. Distance-transitive graphs of valency 3 and 4 were originally classified [2, 11, 12, 13] by using a computer to generate all “feasible intersection arrays” (cf. [1, Chapter 20]). In both cases a classification has since been given by hand [4, 5]. Wecontinue this latter tradition and prove the following theorem—which was recently proved independently by Ivanov et al. using a computer [10].

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Antipodal covering graphs

Cited by 73 publications

References 7 publications

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Imprimitive cometric association schemes: Constructions and analysis

Distance-transitive graphs of valency five

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