The zero-divisor associate graph over a finite commutative ring

Biswas, Bijon; Sen Gupta, Raibatak; Sen, Mridul Kanti; Kar, Sukhendu

doi:10.22049/cco.2023.28488.1577

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	The zero-divisor associate graph over a finite commutative ring
Communications in Combinatorics and Optimization
مقاله 15، دوره 10، شماره 1، خرداد 2025، صفحه 233-244 اصل مقاله (430.38 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2023.28488.1577
نویسندگان
Bijon Biswas¹؛ Raibatak Sen Gupta^* ²؛ Mridul Kanti Sen³؛ Sukhendu Kar⁴
¹Department of Science and Humanities, Ranaghat Government Polytechnic, Nadia - 741201, WB, India
²Department of Mathematics, Bejoy Narayan Mahavidyalaya, West Bengal-712147, India
³Department of Pure Mathematics, University of Calcutta, Kolkata - 700019, India
⁴Department of Mathematics, Jadavpur University, Kolkata - 700032, 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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The zero-divisor associate graph over a finite commutative ring