Sciendo Logo
  1. bookVolume 41 (2021): Issue 2 (November 2021)
Discussiones Mathematicae - General Algebra and Applications's Cover Image
Discussiones Mathematicae - General Algebra and Applications
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
access type Open Access

On Order Prime Divisor Graphs of Finite Groups

Mridul K. Sen,
Mridul K. Sen,
Sunil K. Maity and
Sunil K. Maity and
Sumanta Das
Sumanta Das
Published Online: 06 Sep 2021
Volume & Issue: Volume 41 (2021) - Issue 2 (November 2021)
Page range: 419 - 437
Received: 03 Jul 2019
Accepted: 01 Nov 2020
Discussiones Mathematicae - General Algebra and Applications's Cover Image
Discussiones Mathematicae - General Algebra and Applications
Journal Details
License
Format
Journal
eISSN
2084-0373
First Published
16 Apr 2017
Publication timeframe
2 times per year
Languages
English
Abstract

The order prime divisor graph 𝒫𝒟(G) of a finite group G is a simple graph whose vertex set is G and two vertices a, b ∈ G are adjacent if and only if either ab = e or o(ab) is some prime number, where e is the identity element of the group G and o(x) denotes the order of an element x ∈ G. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph 𝒫𝒟(G) of a group G. Concentrating on the graph 𝒫𝒟(Dn), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of 𝒫𝒟 (ℤn), 𝒫𝒟 (Sn), 𝒫𝒟 (An).

Keywords

  • group
  • dihedral group
  • complete graph
  • Eulerian graph
  • regular graph
  • planar graph
  • order prime divisor graph

MSC 2010

  • 05C25
  • 05C07
  • 05C45
  • 05C76

[1] J. Bosák, The Graphs of Semigroups, in: Theory of Graphs and Application (Academic Press, New York, 1964) 119–125. Search in Google Scholar

[2] B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969) 241–247. https://doi.org/10.21136/CMJ.1969.10089110.21136/CMJ.1969.100891 Search in Google Scholar

[3] I. Chakrabarty, S. Ghosh, T.K. Mukherjee and M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009) 5381–5392. https://doi.org/10.1016/j.disc.2008.11.03410.1016/j.disc.2008.11.034 Search in Google Scholar

[4] I. Chakrabarty, S. Ghosh and M.K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (2009) 410–426. https://doi.org/10.1007/s00233-008-9132-y10.1007/s00233-008-9132-y Search in Google Scholar

[5] D.S. Dummit and R.M. Foote, Abstract Algebra, Third Edition (John Wiley and Sons, Inc., New York, 2004). Search in Google Scholar

[6] A.V. Kelarev and S.J. Quinn, A combinatorial property and power graphs of groups, Contributions to General Algebra 12 (2000) 229–235. Search in Google Scholar

[7] M. Sattanathan and R. Kala, An introduction to order prime graph, Int. J. Contemp. Math. Sciences 4 (2009) 467–474. https://doi.org/10.1111/j.1439-0507.1967.tb02798.x10.1111/j.1439-0507.1967.tb02798.x Search in Google Scholar

[8] D.B. West, Introduction to Graph Theory, Second Edition (Pearson India Education Services Pvt. Ltd., 2017). Search in Google Scholar

[9] B. Zelinka, Intersection graphs of finite Abelian groups, Czech. Math. J. 25 (1975) 171–174. https://doi.org/10.21136/CMJ.1975.10130710.21136/CMJ.1975.101307 Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo