The Complexity of Power Graphs Associated With Finite Groups


  • Steve Kirkland University of Manitoba
  • Ali Reza Moghaddamfar Faculty of Mathematics, K.N. Toosi University of Technology
  • S. Navid Salehy Department of Mathematics, Florida State University
  • S. Nima Salehy Department of Mathematics, Florida State University
  • Mahsa Zohouratar Faculty of Mathematics, K.N. Toosi University of Technology



Power graph, spanning tree, complexity, group.


The power graph P(G) of a finite group G is the graph whose vertex set is
G, with two elements in G being adjacent if one of them is a power of the
other. The purpose of this paper is twofold: (1) to find the complexity of
a clique-replaced graph and study some applications; (2) to derive some
explicit formulas concerning the complexity \kappa(P(G)) for various groups
G such as the cyclic group of order n, the simple groups L_2(q), the extra-
special p-groups of order p^3, the Frobenius groups, etc.


1. J. H. Abawajy, A. V. Kelarev, and M. Chowdhury, Power graphs: a survey, Electronic J. Graph Theory and Applications 1 (2013), 125-147.

2. N. Biggs, Algebraic graph theory, Cambridge University Press, 1974.

3. A. E. Brouwer and W. H. Haemers, Spectra of graphs, Springer, 2012.

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

5. A. Cayley, A theorem on trees, Quart. J. Math. 23 (1889), 376-378.

6. S. Chaiken, A combinatorial proof of the all minors matrix tree theorem, SIAM J. Algebraic Discrete Methods 3 (1982), 319-329.

7. I. Chakrabarty, S. Ghosh, and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (2009), 410-426.

8. B. Huppert, Endliche gruppen i, Springer, 1967.

9. M. I. Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, 2008.

10. A. V. Kelarev, Graph algebras and automata, Marcel Dekker, 2003.

11. A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of groups, Contrib. General Algebra 12 (2000), 229-235.

12. A. V. Kelarev, J. Ryan, and J. Yearwood, Cayley graphs as classifiers for data mining: The influence of asymmetries, Discrete Math. 309 (2009), 5360-5369.

13. Z. Mehranian, A. Gholami, and A. R. Ashrafi, A note on the power graph of a finite group, Int. J. Group Theory 5 (2016), 1-10.

14. R. Merris, Laplacian graph eigenvectors, Linear Algebra Appl. 278 (1998), 221-236.

15. A. R. Moghaddamfar, S. Rahbariyan, S. Navid Salehy, and S. Nima Salehy, The number of spanning trees of power graphs associated with specific groups and some applications, Ars Combinatoria 113 (2017), 269-296.

16. A. R. Moghaddamfar, S. Rahbariyan, and W. J. Shi, Certain properties of the power graph associated with a finite group, J. Algebra Appl. 13 (2014), no. 7, 1450040.