Solvable graphs of finite groups

Year 2020, Volume: 49 Issue: 6, 1955 - 1964, 08.12.2020

Parthajit Bhowal Deiborlang Nongsiang Rajat Nath

https://doi.org/10.15672/hujms.573766

Cited By: 5

Abstract

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $\Gamma_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v \rangle$ is solvable. We show that $\Gamma_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $\Gamma_s (G)$ and derive a lower bound of the clique number of $\Gamma_s (G)$. We prove the non-existence of finite non-solvable groups whose solvable graphs are planar, toroidal, double-toroidal, triple-toroidal or projective. We conclude the paper by obtaining a relation between $\Gamma_s (G)$ and the solvability degree of $G$.

Keywords

solvable graph, genus, solvability degree, finite group

References

• [1] A. Abdollahi, M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra, 38 (12), 4390–4403, 2010.
• [2] A. Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2), 468–492, 2006.
• [3] M. Afkhami, D.G.M. Farrokhi and K. Khashyarmanesh, Planar, toroidal, and projective commuting and non-commuting graphs, Comm. Algebra, 43 (7), 2964–2970, 2015.
• [4] B. Akbari, More on the Non-Solvable Graphs and Solvabilizers, arXiv:1806.01012v1, 2018.
• [5] S. Akbari, A. Mohammadian, H. Radjavi and P. Raja, On the diameters of commuting graphs, Linear Algebra Appl. 418 (1), 161–176, 2006.
• [6] C. Bates, D. Bundy, S. Hart and P. Rowley, A Note on Commuting Graphs for Symmetric Groups, Electron. J. Combin. 16 (1), R6:1–13, 2009.
• [7] J. Battle, F. Harary, Y. Kodama and J.W.T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (6), 565–568, 1962.
• [8] A. Bouchet, Orientable and nonorientable genus of the complete bipartite graph, J. Combin. Theory Ser. B, 24 (1), 24–33, 1978.
• [9] M.R. Darafsheh, H. Bigdely, A. Bahrami and M.D. Monfared, Some results on noncommuting graph of a finite group, Ital. J. Pure Appl. Math. 268, 371–387, 2010.
• [10] A.K. Das and D. Nongsiang, On the genus of the nilpotent graphs of finite groups, Comm. Algebra 43 (12), 5282–5290, 2015.
• [11] A.K. Das, D. Nongsiang, On the genus of the commuting graphs of finite non-abelian groups, Int. Electron. J. Algebra 19, 91–109, 2016.
• [12] S. Dolfi, R.M. Guralnick, M. Herzog and C.E. Praeger, A new solvability criterion for finite groups, J. London Math. Soc. 85 (2), 269–281, 2012.
• [13] J. Dutta and R.K. Nath, Spectrum of commuting graphs of some classes of finite groups, Matematika, 33 (1), 87–95, 2017.
• [14] J. Dutta and R.K. Nath, Finite groups whose commuting graphs integral, Mat. Vesnik, 69 (3), 226–230, 2017.
• [15] J. Dutta and R.K. Nath, Laplacian and signless Laplacian spectrum of commuting graphs of finite groups, Khayyam J. Math. 4 (1), 77–87, 2018.
• [16] P. Dutta, J. Dutta and R.K. Nath, Laplacian spectrum of non-commuting graphs of finite groups, Indian J. Pure Appl. Math. 49 (2), 205–216, 2018.
• [17] H.H. Glover, J.P. Huneke and C.S. Wang, 103 graphs that are irreducible for the projective plane, J. Combin. Theory Ser. B 27 (3), 332–370, 1978.
• [18] R. Guralnick, B. Kunyavskii, E. Plotkin and A. Shalev, Thompson-like characterizations of the solvable radical, J. Algebra, 300 (1), 363–375, 2006.
• [19] R.M. Guralnick and G.R. Robinson, On the commuting probability in finite groups, J. Algebra, 300 (2), 509–528, 2006.
• [20] R. Guralnick and J. Wilson, The probability of generating a finite soluble group, Proc. London Math. Soc. 81 (3), 405–427, 2000.
• [21] D. Hai-Reuven, Non-solvable graph of a finite group and solvabilizers, arXiv:1307.2924v1, 2013.
• [22] R.K. Nath and A.K. Das, On a lower bound of commutativity degree, Rend. Circ. Math. Palermo, 59 (1), 137–141, 2010.
• [23] B.H. Neumann, A problem of Paul Erdös on groups, J. Aust. Math. Soc. (Ser. A), 21 (4), 467–472, 1976.
• [24] D. Nongsiang, Double-Toroidal and Triple-Toroidal Commuting and Nilpotent Graph, Communicated.
• [25] D. Nongsiang and P.K. Saikia, On the non-nilpotent graphs of a group, Int. Electron. J. Algebra, 22, 78–96, 2017.
• [26] A.A. Talebi, On the non-commuting graphs of group $D_{2n}$, Int. J. Algebra, 2 (20), 957–961, 2008.
• [27] D.B. West, Introduction to Graph Theory (Second Edition), PHI Learning Private Limited, New Delhi, 2009.
• [28] A.T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, no. 8., American Elsevier Publishing Co., Inc., New York, 1973.
• [29] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.4, 2013 (http://www.gap-system.org).