The zero-divisor associate graph over a finite commutative ring

Document Type : Original paper

Authors

1 Department of Mathematics, Ranaghat Government Polytechnic, Nadia - 741201, WB, India

2 Department of Mathematics, Bejoy Narayan Mahavidyalaya, P.O.-Itachuna, West Bengal, PIN - 712101

3 Department of Pure Mathematics, University of Calcutta, Kolkata - 700019, India

4 Department of Mathematics, Jadavpur University, India

Abstract

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.

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Main Subjects


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