### Abstract

Let Γ be a G-symmetric graph with vertex set V. We suppose that V admits a G-invariant 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.

Original language | English |
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Pages (from-to) | 403-426 |

Number of pages | 24 |

Journal | Australasian Journal of Combinatorics |

Volume | 71 |

Issue number | 3 |

Publication status | Published - 1 Jan 2018 |

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*Australasian Journal of Combinatorics*,

*71*(3), 403-426.

}

*Australasian Journal of Combinatorics*, vol. 71, no. 3, pp. 403-426.

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

Research output: Contribution to journal › Article

TY - JOUR

T1 - Symmetric graphs with complete quotients

AU - Gardiner, A.

AU - Praeger, Cheryl E.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85046819844&partnerID=8YFLogxK

UR - https://ajc.maths.uq.edu.au/?page=get_volumes&volume=71

M3 - Article

VL - 71

SP - 403

EP - 426

JO - Australasian Journal of Combinatics

JF - Australasian Journal of Combinatics

SN - 1034-4942

IS - 3

ER -