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