# Symmetric Graphs with Large Vertex-Stabilizers

Abstract (Summary)

This thesis deals with trvo cluestions concerning the existence and construction of two fam-
ilies of arc-transitive graphs, all having the property that the number of automorphisms
fixing a vertex is "large" in some sense (which is described in each case).
The first is an infinite family (1") of finite vertex-transitive non-Cayley graphs of fixed
valency, rvith the property that the order of the vertex-stabilizer in a vertex-transitive
group of automorphisms of f' of smallest possible order is a strictly increasing function
of n. This answers a question of Chris Godsil in the affirmative. For each n the graph
f,, is 4-valent and arc-trausitive, with automorphism group a symmetric group of large
prime degree p) 22"+2. The construction uses Sierpinski's gasket to produce generating
permutations for the vertex-stabilizer, which is a large 2-group.
The second is constructed in response to a challenge by Norman Biggs, to produce an
infinite family of 7-arc-transitive quartic graphs with alternating or symmetric automor-
phism groups. It is shown that for all but finitely many positive integers n, there is a finite
connected 7-arc-transitir,'e quartic graph with the alternating group A* acting transitively
on its 7-arcs, and another with the symmetric group ,S,, acting transitively on its 7-arcs.
The proof uses a construction involving permutation representations of a generic finitely-
presented infinite group to obtain finite graphs with the desired property. By a theorem
of Weiss there exist no finite graphs other than simple cycles which are s-arc-transitive
for some s ) 8, hence any finite symmetric graph of degree greater than 2 is at most
7-arc-transitive, and the graphs are constructed to meet this upper bound.
Bibliographical Information:

Advisor:

School:The University of Auckland / Te Whare Wananga o Tamaki Makaurau

School Location:New Zealand

Source Type:Master's Thesis

Keywords:

ISBN:

Date of Publication:01/01/1998