Research Article
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Year 2020, , 1955 - 1964, 08.12.2020
https://doi.org/10.15672/hujms.573766

Abstract

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).

Solvable graphs of finite groups

Year 2020, , 1955 - 1964, 08.12.2020
https://doi.org/10.15672/hujms.573766

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 $G$.

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).
There are 29 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Parthajit Bhowal This is me 0000-0002-8001-9953

Deiborlang Nongsiang This is me 0000-0002-0213-7671

Rajat Nath 0000-0003-4766-6523

Publication Date December 8, 2020
Published in Issue Year 2020

Cite

APA Bhowal, P., Nongsiang, D., & Nath, R. (2020). Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics, 49(6), 1955-1964. https://doi.org/10.15672/hujms.573766
AMA Bhowal P, Nongsiang D, Nath R. Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics. December 2020;49(6):1955-1964. doi:10.15672/hujms.573766
Chicago Bhowal, Parthajit, Deiborlang Nongsiang, and Rajat Nath. “Solvable Graphs of Finite Groups”. Hacettepe Journal of Mathematics and Statistics 49, no. 6 (December 2020): 1955-64. https://doi.org/10.15672/hujms.573766.
EndNote Bhowal P, Nongsiang D, Nath R (December 1, 2020) Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics 49 6 1955–1964.
IEEE P. Bhowal, D. Nongsiang, and R. Nath, “Solvable graphs of finite groups”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 6, pp. 1955–1964, 2020, doi: 10.15672/hujms.573766.
ISNAD Bhowal, Parthajit et al. “Solvable Graphs of Finite Groups”. Hacettepe Journal of Mathematics and Statistics 49/6 (December 2020), 1955-1964. https://doi.org/10.15672/hujms.573766.
JAMA Bhowal P, Nongsiang D, Nath R. Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics. 2020;49:1955–1964.
MLA Bhowal, Parthajit et al. “Solvable Graphs of Finite Groups”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 6, 2020, pp. 1955-64, doi:10.15672/hujms.573766.
Vancouver Bhowal P, Nongsiang D, Nath R. Solvable graphs of finite groups. Hacettepe Journal of Mathematics and Statistics. 2020;49(6):1955-64.