SEMINAR: Groups and Combinatorics Seminar

Groups and Combinatorics Seminar

There will be three 25 minute talks on Tuesday 14th of June in MLR2 starting at 10:30 am.

10:30am:

Sylvia Ozols (Adelaide)

will speak on

The Bruck-Bose Construction

11am:

Wei Jin (UWA)

will speak

On distance, geodesic and arc transitivity of graphs

11:30am

Carmen Amarra (UWA)

will speak on

Quotient-complete arc-transitive graphs



Abstract 1: The Bruck-Bose representation of a projective plane is a tool both for examining objects of any even dimensional projective space in the more familiar setting of a projective plane, and for 'magnifying' objects in certain projective planes by looking at them in a higher dimensional space. In this talk we will go through some finite projective geometry background, including Baer subplanes and partitions of odd dimensional projective spaces. We will define the Bruck-Bose construction and get a general understanding of how it works.



Abstract 2: We compare three transitivity properties of finite graphs, namely, for a positive integer s, s-distance transitivity, s-geodesic transitivity and s-arc transitivity. It is known that if a finite graph is s-arc transitive but not (s+1)-arc transitive then s<8 and s not equal to 6. We show that there are infinitely many geodesic transitive graphs with this property for each of these values of s, and that these graphs can have arbitrarily large diameter if and only if 0< s<4. Moreover, for a prime p we prove that there exists a graph of valency p that is 2-geodesic transitive but not 2-arc transitive if and only if p = 1 (mod 4), and for each such prime there is a unique graph with this property: it is an antipodal double cover of the complete graph K_{p+1} and is geodesic transitive with automorphism group PSL(2,p) x Z_2. This is joint work with A. Devillers, C.H. Li and C. E. Praeger.

Abstract 3: A graph Gamma is G-quotient-complete (for some G in Aut(Gamma)) if it has at least one nontrivial G-normal quotient which is a complete graph, and each of its other nontrivial G-normal quotients is either a complete graph or an empty graph. We define the parameter k to be the number of G-normal quotients of Gamma which are complete, and in this talk we consider the family of quotient-complete graphs with k>2. We construct all the graphs Gamma; in this family together with the corresponding automorphism groups G, and give upper bounds for k in terms of the order of Gamma.