### On the generating graph of a finite group

#### Abstract

_{n}⋊ ℤ

_{m}(direct product ℤ

_{n}× ℤ

_{m}) of cyclic groups ℤ

_{n}and ℤ

_{m}. 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.

#### Keywords

#### Full Text:

PDF#### References

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