Frequency Assignment Model of Zero Divisor Graph

Abstract

Given a frequency assignment network model is a zero divisor graph $\Gamma =\left(V,E\right)$ of commutative ring $R_{\eta }$, 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 $L\left(2,1\right)$ and $L\left(3,2,1\right)$ labeling constraints. For a graph $\Gamma$, $L\left(2,1\right)$ labeling is a nonnegative real valued function $f:V\left(G\right)⟶[0,\infty )$ such that $\mid f\left(x\right)-f\left(y\right)\mid \ge 2d$ if $d=1$ and $\mid f\left(x\right)-f\left(y\right)\mid \ge d$ if $d=2$ where $x$ and $y$ are any two vertices in $V$ and 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 $L\left(h,k\right)$ 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 ${\lambda }_{\left(2,1\right)}$ and ${\lambda }_{\left(3,2,1\right)}$, within ${\Delta }^2$, in terms of maximum and minimum degree of $\Gamma$ by the distance labeling $L\left(2,1\right)$ and $L\left(3,2,1\right)$, respectively, for some order $\eta =p^{n}q,pqr,p^n$ where $p,q,r$ are distinct prime and $n$ is any positive integer.