Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 10 1 2025 03 01 The zero-divisor associate graph over a finite commutative ring 232 243 14655 10.22049/cco.2023.28488.1577 EN Bijon Biswas Department of Science and Humanities, Ranaghat Government Polytechnic, Nadia - 741201, WB, India Raibatak Sen Gupta Department of Mathematics, Bejoy Narayan Mahavidyalaya, West Bengal-712147, India Mridul Kanti Sen Department of Pure Mathematics, University of Calcutta, Kolkata - 700019, India Sukhendu Kar Department of Mathematics, Jadavpur University, Kolkata - 700032, India Journal Article 2023 03 26 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. https://comb-opt.azaruniv.ac.ir/article_14655_1c9367e7e91053c0c9d2b413a822acd7.pdf