Document Type : Original Article

**Author**

Department of Mathematics, Faculty of Sciences, Arak University, Arak, Iran

**Abstract**

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an edge if $\langle x \rangle \subseteq \langle y \rangle$ or $\langle y \rangle \subseteq \langle x \rangle$. Obviously the power graph of any group is always connected, because the identity element of the group is adjacent to all other vertices. We consider $\kappa(G)$, the number of spanning trees of the power graph associated with a finite group $G$. In this paper, for a finite group $G$, first we represent some properties of $\mathcal{P}(G)$, then we are going to find some divisors of $\kappa(G)$, and finally we prove that the simple group $A_6\cong L_2(9)$ is uniquely determined by tree-number of its power graph among all finite simple groups.

**Keywords**

**Main Subjects**

- Abawajy, A. Kelarev, and M. Chowdhury,
*Power graphs: a survey*, Electron. J. Graph Theory Appl. (EJGTA), 1 (2013), pp. 125–147. - Biggs,
*Algebraic graph theory*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, second ed., 1993. - J. Cameron,
*The power graph of a finite group, II*, J. Group Theory, 13 (2010), pp. 779–783. - Cayley,
*A theorem on trees.*, Quart. J., 23 (1888), pp. 376–378. - Chakrabarty, S. Ghosh, and M. K. Sen,
*Undirected power graphs of semigroups*, Semigroup Forum, 78 (2009), pp. 410–426. - Y. Chen, A. R. Moghaddamfar, and M. Zohourattar,
*Some properties of various graphs associated with finite groups*, Algebra Discrete Math., 31 (2021), pp. 195–211. - H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ATLAS
*of finite groups*, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray. - Kelarev,
*Graph algebras and automata*, vol. 257 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 2003. - Kelarev, J. Ryan, and J. Yearwood,
*Cayley graphs as classifiers for data mining: the influence of asymmetries*, Discrete Math., 309 (2009), pp. 5360–5369. - V. Kelarev and S. J. Quinn,
*A combinatorial property and power graphs of groups*, in Contributions to general algebra, 12 (Vienna, 1999), Heyn, Klagenfurt, 2000, pp. 229–235. - Kirkland, A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy, and M. Zohourattar,
*The complexity of power graphs associated with finite groups*, Contrib. Discrete Math., 13 (2018), pp. 124–136. - 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 Combin., 133 (2017), pp. 269– 296. - R. Moghaddamfar, S. Rahbariyan, and W. J. Shi,
*Certain properties of the power graph associated with a finite group*, J. Algebra Appl., 13 (2014), p. 1450040, 18. - S. Rose,
*A course on group theory*, Cambridge University Press, Cambridge-New York-Melbourne, 1978. - N. V. Temperley,
*On the mutual cancellation of cluster integrals in Mayer’s fugacity series*, Proc. Phys. Soc., 83 (1964), pp. 3–16. - B. West,
*Introduction to graph theory*, Prentice Hall, Inc., Upper Saddle River, NJ, second ed., 2001. - V. Zavarnitsine,
*Finite simple groups with narrow prime spectrum*, Sib. Elektron. Mat. Izv., 6 (2009),` pp. 1–12.

April 2024

Pages 81-89