On the generating graph of a finite group

In this paper, we study the generating graph for some finite groups which are semi-direct product ℤn ⋊ ℤm (direct product ℤn × ℤm) of cyclic groups ℤn and ℤm. We show that the generating graphs of them are regular (bi-regular, tri-regular) connected graph with diameter 2 and girth 3 if n and m are prime numbers. Several graph properties are obtained. Furthermore, the probability that 2-randomly elements that generate a finite group G is P (G) = | (a, b) ∈ G×G|G=❬a, b❭|/|G|2. We find the general formula for P (G) of given groups. Our computations are done with the aid of GAP and the YAGs package.