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

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 x y = 0 . In [5], the authors have studied the Laplacian eigenvalues of the graph Γ (Z n) and for distinct proper divisors d 1 , d 2 , ⋯ , d k of n, they defined the sets as, A d i = { x ∈ Z n : (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 ≤ i ≤ k are actually orbits of the group action: A u t (Γ (R)) × R ⟶ R , where A u t (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 × 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. [ABSTRACT FROM AUTHOR]