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 Γ(R) is a simple graph with vertex set as R, and two vertices x, y \in R are adjacent in Γ(R) if and only if $xy = 0$. In [10], the authors have studied the Laplacian eigenvalues of the graph Γ(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(Γ(R)) \times R \longrightarrow R, where Aut(Γ(R)) denotes the automorphism group of Γ(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 Γ(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.