Diameters of Orbital Graphs associated to Groups

Abstract

Associated to groups are a number of graphs, in particular the diameters of orbital graphs associated to primitive groups are of interest. Some work has been performed to describe infinite families of finite primitive permutation groups with a uniform finite upper bound on their orbital graphs. However explicit bounds for smaller families of groups remain a rich area of mathematical research. This paper aims to first build the foundations of group theory before discussing the orbital graphs of primitive groups, before engaging in the direct computation of the diameters of orbital graphs associated to projective special linear groups $\PSL(2,p)$ for primes $p < 200$. Based on the evidence gathered from these computations, we formulate several concrete conjectures.