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

Abstract

Let R be a finite commutative ring with unity, and let G = (V, E) be a simple graph. The zero-divisor graph, denoted by {\Gamma}(R) is a simple graph with vertex set as R, and two vertices x, y \in R are adjacent in {\Gamma}(R) if and only if $xy = 0$. In [10], the authors have studied the Laplacian eigenvalues of 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 show that 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 the automorphism group of {\Gamma}(R). Our main objective is to determine new classes of threshold graphs, since these graphs play an important role in several applied areas. For a reduced ring R, we prove that {\Gamma}(R) is a connected threshold graph if and only if R = F_q or R = F_2 \times F_q. We provide classes of threshold graphs realized by some classes of local rings. Finally, we characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold.