An Upper Bound for the Radio Number of Generalized Gear Graph

Abstract

Let \(d(u,v)\) denote the distance between two distinct vertices of a connected graph \(G\) and \(diam(G)\) be the diameter of \(G\). A radio labeling \(f\) of \(G\) is an assignment of positive integers to the vertices of \(G\) satisfying \(d(u,v) + |f(u) – f(v)| \geq diam(G) + 1\). The maximum integer in the range of the labeling is its span. The radio number of \(G\), denoted by \(rn(G)\), is the minimum possible span. In \([7]\) M. Farooq et al. found the lower bound for the radio number of generalized gear graph. In this paper, we give an upper bound for the radio number of generalized gear graph, which coincides with the lower bound found in \([7]\).