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.