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Let R be a finite commutative ring with unity, and let G = (V, E) be a simplegraph. The zero-divisor graph, denoted by {\Gamma}(R) is a simple graph withvertex set as R, and two vertices x, y \in R are adjacent in {\Gamma}(R) if andonly if $xy = 0$. In [10], the authors have studied the Laplacian eigenvaluesof the graph {\Gamma}(Z_n) and for distinct proper divisors d_1, d_2, \dots,d_k of n, they defined the sets as, A_{d_i} = {x \in Zn : (x, n) = d_i}, where(x, n) denotes the greatest common divisor of x and n. In this paper, we showthat the sets A_{d_i}, 1 \leq i \leq k are actually orbits of the group action:Aut({\Gamma}(R)) \times R \longrightarrow R, where Aut({\Gamma}(R)) denotes theautomorphism group of {\Gamma}(R). Our main objective is to determine newclasses of threshold graphs, since these graphs play an important role inseveral applied areas. For a reduced ring R, we prove that {\Gamma}(R) is aconnected threshold graph if and only if R = F_q or R = F_2 \times F_q. Weprovide classes of threshold graphs realized by some classes of local rings.Finally, we characterize all finite commutative rings with unity of whichzero-divisor graphs are not threshold.

Realization of zero-divisor graphs of finite commutative rings as threshold graphs