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 \( d(u, v) + |f(u) – f(v)| \geq d \), 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.