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The zero-divisor associate graph over a finite commutative ring | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 05 آبان 1402 اصل مقاله (430.94 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28488.1577 | ||
| نویسندگان | ||
| Bijon Biswas1؛ Raibatak Sen Gupta* 2؛ M.K. Sen3؛ Sukhendu Kar4 | ||
| 1Department of Mathematics, Ranaghat Government Polytechnic, Nadia - 741201, WB, India | ||
| 2Department of Mathematics, Bejoy Narayan Mahavidyalaya, P.O.-Itachuna, West Bengal, PIN - 712101 | ||
| 3Department of Pure Mathematics, University of Calcutta, Kolkata - 700019, India | ||
| 4Department of Mathematics, Jadavpur University, India | ||
| چکیده | ||
| In this paper, we introduce the zero-divisor associate graph $\Gamma_D(R)$ over a finite commutative ring $R$. It is a simple undirected graph whose vertex set consists of all non-zero elements of $R$, and two vertices $a, b$ are adjacent if and only if there exist non-zero zero-divisors $z_1, z_2$ in $R$ such that $az_1=bz_2$. We determine the necessary and sufficient conditions for connectedness and completeness of $\Gamma_D(R)$ for a unitary commutative ring $R$. The chromatic number of $\Gamma_D(R)$ is also studied. Next, we characterize the rings $R$ for which $\Gamma_D(R)$ becomes a line graph of some graph. Finally, we give the complete list of graphs with at most 15 vertices which are realizable as $\Gamma_D(R)$, characterizing the associated ring $R$ in each case. | ||
| کلیدواژه‌ها | ||
| Zero-divisor؛ Commutative ring؛ Chromatic number؛ Complete graph؛ Line graph | ||
| مراجع | ||
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