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.