Solution to a Question on a Family of Imprimitive Symmetric Graphs

Xu, Guangjun; Zhou, Sanming

doi:10.1017/s0004972710000262

Cited by 5 publications

(3 citation statements)

References 7 publications

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“…If there is such a quotient, what information does this give us about the original graph? These questions were studied in [7,8,10,11,13,14,16,17], with a focus on the case where v − k ≥ 1 or k ≥ 1 is small. In the present paper we consider the more general case where k = v − p for a prime p ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Graphs With 2-Arc Transitive Quotients

Zhou

2014

J. Aust. Math. Soc.

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A graph Γ is G-symmetric if Γ admits G as a group of automorphisms acting transitively on the set of vertices and the set of arcs of Γ, where an arc is an ordered pair of adjacent vertices. In the case when G is imprimitive on V (Γ), namely when V (Γ) admits a nontrivial G-invariant partition B, the quotient graph Γ B of Γ with respect to B is always G-symmetric and sometimes even (G, 2)-arc transitive. (A G-symmetric graph is (G, 2)-arc transitive if G is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for Γ B to be (G, 2)-arc transitive (regardless of whether Γ is (G, 2)-arc transitive) in the case when v − k is an odd prime p, where v is the block size of B and k is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of v, k and two other parameters with respect to (Γ, B) together with a certain 2-point transitive block design induced by (Γ, B). We prove further that if p = 3 or 5 then these necessary conditions are essentially sufficient for Γ B to be (G, 2)-arc transitive.

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Section: Introductionmentioning

confidence: 99%

Symmetric Graphs With 2-Arc Transitive Quotients

Zhou

2014

J. Aust. Math. Soc.

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“…The paper [6] studies varous properties of these graphs, and then focuses on the situations where Γ B is a complete graph or a cycle, and the cases where k is 1, 2, v − 1 or v, where v = |B i |. Many of these special cases (and also the case k = v − 2 [15,18]) are explored further in the literature, especially by Sanming Zhou and his co-authors. In particular they classified all triples (Γ, G, B) such that Γ B is a complete graph and k = v − 1, a culmination of work in [2,4,10,19]; and they studied more general quotients Γ B , especially the case where G acts 2-arc transitively on Γ B , see [12,14,16,20,21,22].…”

Section: Introductionmentioning

confidence: 99%

Symmetric graphs with complete quotients

Gardiner,

Praeger

2014

Preprint

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Let Γ be a G-symmetric graph with vertex set V . We suppose that V admits a Ginvariant partition B = {B = B 0 , B 1 , . . . , B b }, with parts B i of size v, and that the quotient graph Γ B induced on B is a complete graph K b+1 . Then, for each pair of suffices i, j (i = j), the graph B i , B j induced on B i ∪ B j is bipartite with each vertex of valency 0 or t (a constant). When t = 1, it was shown earlier how a flag-transitive 1-design D(B) induced on the part B can sometimes be used to classify possible triples (Γ, G, B). Here we extend these ideas to t ≥ 1 and prove that, if G(B) B is 2-transitive and the blocks of D(B) have size less than v, then either (i) v < b, or (ii) the triple (Γ, G, B) is known explicitly.

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“…Graphs associated with various algebraic structures have been actively investigated and many interesting results have recently been obtained, for example in [14][15][16]19]. An extensive bibliography concerning applications of graphs of this kind is given in [12]; see also [8].…”

Section: Introductionmentioning

confidence: 99%