On the radio antipodal geometric mean number of ladder related graphs

Abstract

Let $G(V, E)$ be a graph with vertex set $V$ and edge set $E$. A radio geometric mean labeling of a connected graph $G$ is a one to one map from the vertex set $V(G)$ to the set of natural numbers $N$ such that for two distinct vertices $u$ and $v$ of $G$, $d(u,v)+\lceil \sqrt{f(u)f(v)} \rceil \geq 1 + diam(G)$, where $d(u, v)$ represents the shortest distance between the vertices $u$ and $v$ and $diam(G)$ represents the diameter of $G$ . Based on the concept of radio geometric mean labeling, a new graph labeling called \textit{radio antipodal geometric mean labeling} is being introduced in this paper. A radio antipodal geometric mean labeling of a graph $G$ is a mapping from the vertex set $V(G)$ to the set of natural numbers $N$ such that for two distinct vertices $u$ and $v$ of $G$, $d(u,v) + \lceil \sqrt{f(u)f(v)} \rceil \geq diam(G)$. If $d(u, v) = diam(G)$, then the vertices $u$ and $v$ can be given the same label and if $d(u, v) \neq diam(G)$ then the vertices $u$ and $v$ should be assigned different labels. The radio antipodal geometric mean number of $f$, $r_{agmn}(f)$ is the maximum number assigned to any vertex of $G$. The radio antipodal geometric mean number of $G$, $r_{agmn}(G)$ is the minimum value taken over all radio antipodal geometric mean labeling $f$ of $G$. In this paper, the radio antipodal geometric mean number of certain ladder related graphs have been investigated.