Raziskovalni matematični seminar - Arhiv

This talk gives a survey on the state of the art in the classification of the finite groups of prime-power order. Given a prime p and a natural number n, one can consider the function f(p,n) of groups of order p^n (up to isomorphism). The function f(p,n) can be considered as a function in p or as a function in n and then has different interesting asymptotic properties. However, using the order is not the only way to approach the classification of groups of prime-power order. An alternative approach is to classify the groups of prime-power order using their coclass as primary invariant. The talk will explain this in more detail and also exhibit some of the results of this approach.

We consider the problem of finding the smallest vertex-transitive k-regular graphs of girth g. Counting cycles and using number theory's theorems we extend Biggs's result proving that the asymptotic density of the set of all odd girths g for which exists a vertex-transitive (k,g)-cage with excess not exceeding e is 0. Also, we consider the existence of (k,g)-vertex-transitive graphs when g is even and the excess is 4 or smaller than min{k-2,g}.

Based on joint work with Robert Jajcay.

The family of generalized Petersen graphs G(n,k), introduced by Coxter et al. (1950) and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover of a simple undirected graph G  is a a special type of bipartite covering graph of G, isomorphic to the direct (tensor) product of G and K_{2}. In the seminar we will identify generalized Petersen graphs G(n,k) that are Kronecker covers of another generalized Petersen graphs, and discuss some related questions.