Generalized Commuting Graph of Dihedral, Semi-dihedral and Quasi-dihedral Groups

Authors

  • Mustafa Anis El-Sanfaz Department of Mathematics, Statistics and Physics, College of Arts, Qatar University, Doha, Qatar
  • Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Siti Norziahidayu Amzee Zamri UniSZA Science and Medicine Foundation Centre, Universiti Sultan Zainal Abidin, Gong Badak Campus, 21300 Kuala Nerus, Terengganu

DOI:

https://doi.org/10.11113/mjfas.v17n6.2245

Keywords:

graph theory, commuting graph, dihedral groups, semi-dihedral groups, quasi-dihedral groups

Abstract

Commuting graphs are characterized by vertices that are non-central elements of a group where two vertices are adjacent when they commute. In this paper, the concept of commuting graph is extended by defining the generalized commuting graph. Furthermore, the generalized commuting graph of the dihedral groups, the quasi-dihedral groups and the semi-dihedral groups are presented and discussed. The graph properties including chromatic and clique numbers are also explored.

References

J. Bondy and G. Murty, 1982. Graph Theory with Application, 5th ed. North Holland, Boston, New York.

C. Godsil and G. Royle. 2001. Algebraic Graph Theory, 5th ed. Springer, Boston New York.

R. Brauer, and K. Fowler, “On groups of even order,” Ann Math, vol. 62, no. 3, pp. 565-583, 1955.

G. S. Singh, 2010. Graph Theory, PHI Private Learning Limited, New Delhi.

G. Chartrand, L. Lesniak and P. Zhang, 2010. Graphs and Diagraphs, CRC Press, Boca Raton.

Y. Segev, “The commuting graph of minimal nonsolvable groups,” Geomtriae Dedicata, vol. 88, pp. 55-66, 2001.

D. Bundy, “The connectivity of commuting graphs,” Journal of Combinatorial Theory, Series A, vol. 113, no. 9951007, 2006.

A. Iranmanesh, and A. Jafarzadeh, “On the commuting graph associated with the symmetric and alternating groups,” Journal of Algebra and Its Applications, vol. 7, no. 1, 129146, 2008.

C. Bates, D. Bundy, S. Hart, and P. Rowley, “A note on commuting graphs for symmetric groups,” The Electronic Journal of Combinatorics, vol. 16, no. 1, 2009.

T. Chelvam, and L. R. S. Selvakumar, “Commuting graphs on dihedral group,” Journal of Mathematics and Computer Science, vol. 2, no. 2, pp. 402-406, 2011.

R. Raza, and S. Faizi, “Commuting graphs of dihedral types groups,” Applied Mathematics E-Notes, vol. 13, pp. 221-227, 2013.

C. Parker, “The commuting graph of soluble,” Bulleting of London Mathematical Society, vol. 45, no. 4, pp. 839-848, 2013.

A. Azimi, A. Erfanian, and M. D. G. Farrokhi, “The Jacobson graph of commutative rings,” Journal of Algebra and Its Applications, vol. 12, no. 3, pp. 1250179, 2013.

A. Azimi, A. Erfanian, and M. D. G. Farrokhi, “Isomorphism between Jacobson graphs,” Rendiconti del Circolo Matematico di Palermo, vol. 63, no. 2, pp. 277-286, 2014.

A. Azimi, and M. D. G. Farrokhi, “Cycles and paths in Jacobson graphs,” Ars Combinatoria, vol. 134, pp. 61-74, 2017.

H. Ghayour, A. Erfanian, and A. Azimi, “Some results on the Jacobson graph of a commutative ring,” Rendiconti del Circolo Matematico di Palermo Series 2, vol. 67, no. 1, pp. 33-41, 2018.

S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, “The inclusion ideal graph of rings,” Communications in Algebra, vol. 43, no. 6, pp. 2457-2465, 2015.

A. Das, and D. Nongsiang, “On the genus of the commuting graphs of finite non-abelian groups,” International Electronic Journal of Algebra, vol. 19, pp. 91-109, 2016.

N. Zaid, N. H. Sarmin and S. N. A. Zamri, “A Variant of Commutativity Degree and Its Generalized Conjugacy Class Graph,” Advanced Science Letters, vol. 24, no. 6, pp. 4429-4432, 2018.

N. Zaid, N. H. Sarmin and S. M. S. Khasraw, “On the probability and graph of some finite rings of matrices,” AIP Conference Proceedings, vol. 2266, pp. 060010, 2020.

N. Zaid, N. H. Sarmin and S. M. S. Khasraw, “Probabilistic characterizations of some finite rings of matrices and its zero divisor graph,” Southeast Asian Bulletin of Mathematics, vol. 44, no. 6, pp. 859-866, 2020.

V. V. Swathi and M. S. Sunitha, “Square graphs of finite groups,” AIP Conference Proceedings, vol. 2336, pp. 050014, 2021.

M. A. El-Sanfaz, N. H. Sarmin, and S. M. S. Omer, “On the probability that a group element of the dihedral groups fixes a set,” International Journal of Applied Mathematics and Statistics, vol. 52, no. 1, pp. 1-6, 2014.

J. A. Gallian. 2002. Contemporary Abstract Algebra, 5th ed. Houghton Mifflin Company, Boston, New York,

M. A. El-Sanfaz, N. H. Sarmin, and S. M. S. Omer, “On the probability that a group element fixes a set and its generalized conjugacy class graph,” International Journal of Mathematical Analysis, vol. 9, no. 4, pp. 161-167, 2015.

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Published

31-12-2021