The zero-divisor associate graph over a finite commutative ring

In this paper, we introduce the zero-divisor associate graph ΓD(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 z1, z2 in R such that az1 = bz2. We determine the necessary and sufficient conditions for connectedness and completeness of ΓD(R) for a unitary commutative ring R. The chromatic number of ΓD(R) is also studied. Next, we characterize the rings R for which ΓD(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 ΓD(R), characterizing the associated ring R in each case.