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.
Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0and only if G can be obtained from the code by some rule. Some classes of graphs such as threshold and chain graphs realizes a code of the type mentioned above. The main objective of this research article is to develop some computationally feasible methods to determine some interesting graph theoretical invariants. We present an efficient algorithm to determine the metric dimension of threshold and chain graphs. We compute threshold dimension and restricted threshold dimension of threshold graphs. We discuss L(2, 1)-coloring in threshold and chain graphs. In fact, for every threshold graph G, we establish a formula by which we can obtain the λ-chromatic number of G. Finally, we provide an algorithm to compute the λ-chromatic number of chain graphs.
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