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.
School Location:New Zealand
Source Type:Master's Thesis
Date of Publication:01/01/1998