Frequency Assignment Model of Zero Divisor Graph

Given a frequency assignment network model is a zero divisor graph Γ = V , E of commutative ring R η , in this model, each node is considered to be a channel and their labelings are said to be the frequencies, which are assigned by the L 2 , 1 and L 3 , 2 , 1 labeling constraints. For a graph Γ , L 2 , 1 labeling is a nonnegative real valued function f : V G ⟶ 0 , ∞ such that ∣ f x − f y ∣ ≥ 2 d if d = 1 and ∣ f x − f y ∣ ≥ d if d = 2 where x and y are any two vertices in V and d > 0 is a distance between x and y . Similarly, one can extend this distance labeling terminology up to the diameter of a graph in order to enhance the channel clarity and to prevent the overlapping of signal produced with the minimum resource (frequency) provided. In general, this terminology is known as the L h , k labeling where h is the difference of any two vertex frequencies connected by a two length path. In this paper, our aim is to find the minimum spanning sharp upper frequency bound λ 2 , 1 and λ 3 , 2 , 1 , within Δ 2 , in terms of maximum and minimum degree of Γ by the distance labeling L 2 , 1 and L 3 , 2 , 1 , respectively, for some order η = p n q , p q r , p n where p , q , r are distinct prime and n is any positive integer.