Radio Antipodal Number of Gird like Architecture Graphs

Abstract

Let $G=(V,E)$ be a graph with vertex set $V$ and edge set $E$. Let $\text{diam}(G)$ denote the diameter of $G$ and $d(u,v)$ denote the distance between the vertices $u$ and $v$ in $G$. An antipodal labeling of $G$ with diameter $d$ is a function $f$ that assigns to each vertex $u$, a positive integer $f(u)$, such that $d(u,v)+|f(u)–f(v)|\ge d$, for all $u,v\in V$. The span of an antipodal labeling $f$ is $max\{|f(u)–f(v)|:u,v\in V(G)\}$. The antipodal number for $G$, denoted by $\text{an}(G)$, is the minimum span of all antipodal labelings of $G$. Determining the antipodal number of a graph $G$ is an NP-complete problem. In this paper, we determine the antipodal number of certain graphs.