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Abstract
We introduce the non-braid graph of a group G, denoted by ζ(G), as a graph with vertex set G \ B(G), where B(G) is the braider of G, defined as the set {x ∈ G | (∀y ∈ G)xyx = yxy}, and two distinct vertices x and y are joined by an edge if and only if xyx ̸ = yxy. In this paper particularly we give the independent number, the vertex chromatic number, the clique number, and the minimum vertex cover of non-braid graph of dihedral group Dn
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References
- Abdollahi, A., Akbari, S., Maimani, H.R., 2006, Non-commuting graph of a group, Journal of Algebra, 298, 468–492.
- Cayley, A., 1878, Desiderata and suggestions: No. 2. The theory of groups: Graphical representation, Am. J. Math., 1, 174–176.
- Chartrand, Gary and Zhang, P., 2005 Introduction to Graph Theory, New York: McGraw Hill.
- Chen, J., Kou, L., Cui, X., 2016, Approximation Algorithm for the Minimum Vertex Cover Problem, Procedia Engineering, 137:180-185.
- Rahayuningtyas, H., Abdussakir, and Nashichuddin, A 2015, Bilangan kromatik graf commuting dan non commuting grup dihedral, CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 4, no. 1, 16 -21.
- Talebi, A. A., 2015, On the Non-Commuting Graphs of Group D2n, International Journal of Algebra, Vol 2. No. 20, 957 - 961.
- Wilson, R. J., 1996, Introduction to Graph Theory, Fourth Edition, England: Addison Wesley Longman Limited.
- Cahyati, S. E., Fadhiilah, A. R., Candra, D. A., Wijayanti, E. I., 2022, Non-Braid Graphs of Ring Zn, Jurnal Teori dan Aplikasi Matematika, Vol 6. No. 1, 106-116.
- Hubbi Muhammad, Niswah Qonita, R.A.W. Fibrianti, Yeni Susanti, 2023, Bipartite Graph Associated With Elements and Cosets of Subrings of Finite Rings, BAREKENG: Journal of Mathematics and Its Application, vol. 17, no. 2, 0667-0672.
References
Abdollahi, A., Akbari, S., Maimani, H.R., 2006, Non-commuting graph of a group, Journal of Algebra, 298, 468–492.
Cayley, A., 1878, Desiderata and suggestions: No. 2. The theory of groups: Graphical representation, Am. J. Math., 1, 174–176.
Chartrand, Gary and Zhang, P., 2005 Introduction to Graph Theory, New York: McGraw Hill.
Chen, J., Kou, L., Cui, X., 2016, Approximation Algorithm for the Minimum Vertex Cover Problem, Procedia Engineering, 137:180-185.
Rahayuningtyas, H., Abdussakir, and Nashichuddin, A 2015, Bilangan kromatik graf commuting dan non commuting grup dihedral, CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 4, no. 1, 16 -21.
Talebi, A. A., 2015, On the Non-Commuting Graphs of Group D2n, International Journal of Algebra, Vol 2. No. 20, 957 - 961.
Wilson, R. J., 1996, Introduction to Graph Theory, Fourth Edition, England: Addison Wesley Longman Limited.
Cahyati, S. E., Fadhiilah, A. R., Candra, D. A., Wijayanti, E. I., 2022, Non-Braid Graphs of Ring Zn, Jurnal Teori dan Aplikasi Matematika, Vol 6. No. 1, 106-116.
Hubbi Muhammad, Niswah Qonita, R.A.W. Fibrianti, Yeni Susanti, 2023, Bipartite Graph Associated With Elements and Cosets of Subrings of Finite Rings, BAREKENG: Journal of Mathematics and Its Application, vol. 17, no. 2, 0667-0672.