Document Type : Research Paper

**Authors**

Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran

**Abstract**

Let $G$ be a finite group which is not cyclic of prime power order. The join graph $\Delta(G)$ of $G$ is a graph whose vertex set is the set of all proper subgroups of $G$, which are not contained in the Frattini subgroup $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $G=\langle H, K\rangle$. Among other results, we show that if $G$ is a finite cyclic group and $H$ is a finite group such that $\Delta(G)\cong\Delta(H)$, then $H$ is cyclic. Also we prove that $\Delta(G)\cong\Delta(A_5)$ if and only if $G\cong A_5$.

**Keywords**

[1] H. Ahmadi and B. Taeri, A graph related to the join of subgroups of a finite group, Rend. Sem. Math., 131 (2014)

281–292.

281–292.

[2] H. Ahmadi and B. Taeri, On the planarity of a graph related to the join of subgroups of a finite group, Bull. Iranian

Math. Soc., 40 (2014) 1413–1431.

Math. Soc., 40 (2014) 1413–1431.

[3] H. Ahmadi and B. Taeri, Finite groups with regular join graph of subgroups, J. Algebra Appl., 15 (2016) 10 pages.

[4] Z. Bahrami and B. Taeri, Further results on the join graph of a finite group, Turk. J. Math., 43 (2019) 2097–2113.

[5] A. Ballester-Bolinches, J. Cossey and R. Esteban-Romero, On a graph related to permutability in finite groups,

Ann. Mat. Pura Appl., 189 (2010) 567–570.

Ann. Mat. Pura Appl., 189 (2010) 567–570.

[6] Bos´ak J, The graphs of semigroups in theory of graphs and applications, Proceedings of the Symposium on theory

of graphs, and its applications., (1963) 119–125.

of graphs, and its applications., (1963) 119–125.

[7] T. Breuer, R. M. Guralnick, A. Lucchini, A. Mar´oti and G. P. Nagy, Hamiltonian cycles in the generating graphs

of finite groups, Bull. Lond. Math. Soc., 42 (2010) 621–633.

of finite groups, Bull. Lond. Math. Soc., 42 (2010) 621–633.

[8] J. R. Britnell, A. Evseev, R. M. Guralnick, P. E. Holmes and A. Mar´oti, Sets of that pairwise generate a linear

group, J. Combin. Theory Ser. A., 115 (2008) 442–465.

group, J. Combin. Theory Ser. A., 115 (2008) 442–465.

[9] I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math.,

309 (2009) 5381–5392.

309 (2009) 5381–5392.

[10] B. Cs´ak´any and G. Poll´ak, The graph of subgroups of a finite group, Czechoslovak Math. J., 19 (1969) 241–247.

[11] The GAP Group, GAP-Groups, Algorithms and Programing, Version 4.4; (2005), http://www.gap-system.org.

[12] M. Herzog, P. Longobardi and M. Maj, On a graph related to subgroups of a group, Bull. Aust. Math. Soc., 81

(2010) 317–328.

(2010) 317–328.

[13] A. Lucchini and A. Mar´oti, Some results and questions related to the generating graph a finite group, Ischia group

theory 2008 (ed. M. Bianchi et al.), Proceedings of the Conference in Group Theory, Naples, Italy, April 1-4, 2008

(World Scientific, 2009) 183–208.

theory 2008 (ed. M. Bianchi et al.), Proceedings of the Conference in Group Theory, Naples, Italy, April 1-4, 2008

(World Scientific, 2009) 183–208.

[14] A. Lucchini and A. Mar´oti, On the clique number of the generating graph of a finite group, Proc. Amer. Math. Soc.,

137 (2009) 3207–3217.

137 (2009) 3207–3217.

[15] A. Lucchini and A. Mar´oti, On finite simple groups and Kneser graphs, J. Algebraic Combin., 30 (2009) 549–566.

[16] A. Lucchini, The diameter of the generating graph of a finite soluble group, J. Algebra, 492 (2017) 28–43.

[17] S. Pemmaraju and S. Skiena, Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, Cambridge University Press, 2003.

[18] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag New York Heidelberg Berlin, 1996.

[19] P. D. Sharma and S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra, 176 (1995) 124–127.

[20] B. Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Math. J., 25 (1975) 171–174.

[21] D. B. West, Introduction to graph theory, Second Ed., Pearson Education, Inc., 2001.

December 2021

Pages 175-186

**Receive Date:**31 May 2020**Revise Date:**04 October 2020**Accept Date:**05 October 2020**Published Online:**01 December 2021