# Symmetric graphs with complete quotients

A. Gardiner, Cheryl E. Praeger

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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 language English 403-426 24 Australasian Journal of Combinatorics 71 3 Published - 1 Jan 2018

### Fingerprint

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

### Cite this

@article{67a4377a500e4afea1d25b1d302cc8f3,
title = "Symmetric graphs with complete quotients",
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.",
author = "A. Gardiner and Praeger, {Cheryl E.}",
year = "2018",
month = "1",
day = "1",
language = "English",
volume = "71",
pages = "403--426",
journal = "Australasian Journal of Combinatics",
issn = "1034-4942",
publisher = "The University of Queensland Press",
number = "3",

}

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

Research output: Contribution to journalArticle

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 -