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 .