Symmetric graphs with complete quotients

A. Gardiner, Cheryl E. Praeger

Research output: Contribution to journalArticle

1 Citation (Scopus)
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Abstract

Let Γ be a G-symmetric graph with vertex set V. We suppose that V admits a G-invariant partition B = {B = B0, B1, …, Bb}, with parts Bi of size v, and that the quotient graph ΓB induced on B is a complete graph Kb+1. Then, for each pair of suffices i, j (i ≠ j), the graph 〈Bi, Bj 〉induced on Bi ∪ Bj 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.

Original languageEnglish
Pages (from-to)403-426
Number of pages24
JournalAustralasian Journal of Combinatorics
Volume71
Issue number3
Publication statusPublished - 1 Jan 2018

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Symmetric Graph
Quotient
Quotient Graph
Flag-transitive
Vertex of a graph
Complete Graph
Classify
Partition
Invariant
Graph in graph theory
Design

Cite this

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Symmetric graphs with complete quotients. / Gardiner, A.; Praeger, Cheryl E.

In: Australasian Journal of Combinatorics, Vol. 71, No. 3, 01.01.2018, p. 403-426.

Research output: Contribution to journalArticle

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AB - Let Γ be a G-symmetric graph with vertex set V. We suppose that V admits a G-invariant partition B = {B = B0, B1, …, Bb}, with parts Bi of size v, and that the quotient graph ΓB induced on B is a complete graph Kb+1. Then, for each pair of suffices i, j (i ≠ j), the graph 〈Bi, Bj 〉induced on Bi ∪ Bj 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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