### 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:

PDFDOI: http://dx.doi.org/10.19184/ijc.2024.8.1.3

#### References

E. Bertram, M. Herzog, and A. Mann, On a graph related to conjugacy classes of groups, Bulletin of the London Mathematical Society, **22**(6), (1990), 569–575.

T. Breuer, R. Guralnick, and W. Kantor, Probabilistic generation of finite simple groups, *II Journal of Algebra*, **320**(2), (2008), 443–494.

A. Erfanian and B. Tolue, Conjugate graphs of finite groups, *Discrete Mathematics, Algorithms and Applications*, **4**(2) (2012), 1250035.

B. Esther, Probability of generating a dicyclic group using two elements Pi Mu Epsilon Journal, **14**(3), (2015), 165–168.

M. W. Liebeck and A. Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, *J. Algebra*, **184** (1996), 31–57.

K. L. Patti, The probability of randomly generating a finite group Pi Mu Epsilon Journal, **11** (6), (2002), 313–316.

H. Tong-Viet, Finite groups whose prime graphs are regular, *Journal of Algebra*, **397** (2014), 18–31.

R. J. Wilson, Introduction to graph theory, *Pearson Education India*, (1979).

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