Document Type : Research Paper

**Authors**

Department of Mathematics, Science and Research Branch Islamic Azad University, Tehran, Iran

**Abstract**

For a given odd prime $p$, we investigate the power graphs of three classes of finite groups: the elementary abelian groups of exponent $p$, and the extra special groups of exponents $p$ or $p^2$. We show that these power graphs are Eulerian for every $p$. As a corollary, we describe two classes of non-isomorphic groups with isomorphic power graphs. In addition, we prove that the clique graphs of the power graphs of two considered classes are complete.

**Keywords**

**Main Subjects**

[1] J. Abawajy, A. V. Kelarev and M. Chowdhury, Power graphs: a survey, Electronic J. Graph Theory and Applications,

1 (2013) 125–147.

1 (2013) 125–147.

[2] M. Akbari and A. Moghaddamfar, Groups for which the noncommuting graph is a split graph, Int. J. Group Theory,

6 no. 1 (2017) 29–35.

6 no. 1 (2017) 29–35.

[3] A. R. Ashrafi and B. Soleimani, Normal edge-transitive and 12-arc-transitive Cayley graphs on non-abelian groups

of order 2pq, p > q are odd primes, Int. J. Group Theory, 3 no. 5 (2016) 1–8.

of order 2pq, p > q are odd primes, Int. J. Group Theory, 3 no. 5 (2016) 1–8.

[4] P. J. Cameron, The power graph of a finite group II, J. Group Theory, 13 (2010) 779–783.

[5] P. J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Math., 311 (2011) 1220–1222.

[6] I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum., 78 (2009)

410–426.

410–426.

[7] T. T. Chelvam and M. Sattanathan, Power graphs of finite abelian groups, Algebra and Discrete math., 16 (2013)

33–41.

33–41.

[8] H. R. Dorbidi, A note on the coprime graph of a group, Int. J. Group Theory, 5 no.4 (2016) 17–22.

[9] R. Hafezieh, On nonsolvable groups whose prime degree graphs have four vertices and one triangle, Int. J. Group

Theory, 6 no. 3 (2017) 1–6.

Theory, 6 no. 3 (2017) 1–6.

[10] R. Hafezieh, Bipartite divisor graph for the set of irreducible character degrees, Int. J. Group Theory, 6 no. 4 (2017)

41–51.

41–51.

[11] D. L. Johnson, Presentation of groups, Second Ed, London Math. Soc. Student Texts, Cambridge University Press,

Cambridge, 1997.

Cambridge, 1997.

[12] A. Kelarev, J. Ryan and J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries,

Discrete Math., 309 (2009) 5360–5369.

Discrete Math., 309 (2009) 5360–5369.

[13] A. V. Kelarev, Ring constructions and applications, World Scientific, NJ, 2002.

[14] A. V. Kelarev, Graph algebras and automata, M. Dekker, New York, 2003.

[15] A. Mehranian, A. R. Gholami and A. Ashrafi, Note on the power graph of a finite group, Int. J. Group Theory, 5

no. 1 (2016) 1–10.

no. 1 (2016) 1–10.

[16] S. A. Moosavi, On bipartite divisor graph for character degrees, Int. J. Group Theory, 6 no. 1 (2017) 1–7.

[17] G. B. Preston, Semidirect product of semigroups, Proc. Royal Soc. Edinburgh, 102 (A) (1986) 91–102.

[18] E. F. Robertson and Y. Unl¨u, On semigroup presentations, ¨ Proc. Edinburg Math. Soc., 36 (1992) 55–68.

[19] M. Shaker and M. A. Iranmanesh, On groups with specified quotient power graphs, Int. J. Group Theory, 5 (2016)

49–60.

49–60.

[20] T. Tamura, Examples of direct product of semigroups or groupoids, Notices of the American Math. Soc., 8 (1961)

419–422.

419–422.