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Non-identity order divisor graphs of groups | ||
| Journal of Algebra and Related Topics | ||
| دوره 14، شماره 1، آبان 2026، صفحه 199-206 اصل مقاله (159.18 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.28373.1716 | ||
| نویسندگان | ||
| M. Sattanathan1؛ J. Kottarathil* 2؛ I. Chakrabarty3 | ||
| 1Department of Mathematics, Sri Paramakalyani College, Alwarkurichi,Tamil Nadu, India | ||
| 2Department of Mathematics, St. Joseph’s College (Autonomous), Devagiri, Kozhikode, Kerala, India | ||
| 3Department of Mathematics, CHRIST(Deemed to be University), Bengaluru, India | ||
| چکیده | ||
| Let $G$ be a group with identity $e$. In this paper, we define and study the non-identity order divisor graph of $G$, where the vertex set is $G-\{e\}$ and two distinct vertices $x$ and $y$ are adjacent if and only if either $O(x)|O(y)$ or $O(y)|O(x)$. We denote the order divisor graph of group $G$ by $\o(G)$. We study some basic properties of $\o(G)$ such as connectedness, completeness, bipartiteness and Eulerian property. The lower bound as well as the number of edges of $\o(G)$ are also calculated for some group $G$ and some characterizations for fundamental properties of $\o(G)$ have been obtained. Finally, we explore the relation between the order prime graph and the non-identity order divisor graph of some group $G$. | ||
| کلیدواژهها | ||
| Non-identity order divisor graph؛ Order prime graph؛ Eulerian graph؛ Finite group | ||
| مراجع | ||
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[1] A. Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, J. Algebra, (2) 298 (2006), 468-492. [2] S. Akbari and A. Mohammadian, On the zero-divisor graph of commutative ring, J. Algebra, (2) 274 (2004), 847-855. [3] T. Chalapathi and R.V.M.S.S. KiranKumar, Order divisor graphs of finite groups, Malaya J. Mat., (2) 5 (2017), 464-474. [4] G. Chartrand and P. Zhang, Introduction to graph theory, Tata McGraw-Hill, 2006. [5] J. A. Gallian, Contemporary abstract algebra, Narosa Publishing House, 1999. [6] S. U. Rehman, A. Q. Baig, M. Imran and Z.U. Khan, Order divisor graphs of finite groups, An. Stiint. Univ. Ovidius Constanta Ser. Mat., (3) 26 (2018), 29-40. [7] M. Sattanathan and R. Kala, An introduction to order prime graph, Int. J. Contemp. Math. Sci., (10) 4 (2009), 467-474. [8] M. Sattanathan, J. Kottarathil and R.M. Muthu Lakshmi, 2-order prime graph, under review (2024). [9] M. K. Sen, S. K. Maity and S. Das, On order prime divisor graphs of finite groups, Discuss. Math. Gen. Algebra Appl., (2) 41 (2021), 419-437. [10] T. Tamizh Chelvam and M. Sattanathan, Subgroup intersection graph of a group, J. Adv. Research in Pure Math., (4) 3 (2011), 44-49. [11] T. Tamizh Chelvam and M. Sattanathan, Subgroup intersection graph of finite abelian groups, Trans. Comb., (3) 1 (2012), 5-10. | ||
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