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.