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 ∈ R are adjacent in Γ(R) if and only if xy = 0. In [6], the authors have studied the Laplacian eigen values of the graph Γ(Zn) and for distinct proper divisors d1, d2, • • • , d k of n, they defined the sets aswhere (x, n) denotes the greatest common divisor of x and n. In this paper, we show that the sets A d i , where 1 ≤ i ≤ k are actually the orbits of the group action: Aut(Γ(R)) × R −→ R, where Aut(Γ(R)) denotes the automorphism group of Γ(R). We characterize all finite commutative rings with unity of which zero-divisor graphs are not threshold. We study creation sequences, hyperenergeticity and hypoenergeticity of zero-divisor graphs. We compute the Laplacian eigenvalues of zero-divisor graphs realized by some classes of reduced and local rings. We show that the Laplacian eigenvalues of zero-divisor graphs are the representatives of orbits of the group action:Aut(Γ(R)) × R −→ R.