Radial Radio Number of Hexagonal and Its Derived Networks

Abstract

A mapping $\mathrm{\daleth }:V\left(G\right)⟶N{\displaystyle \cup }\left\{0\right\}$ for a connected graph $G=\left(V,E\right)$ is called a radial radio labelling if it satisfies the inequality $\left|\mathrm{\daleth }\left(x\right)-\mathrm{\daleth }\left(y\right)\right|+d\left(x,y\right)\ge \text{rad}\left(G\right)+1$ $\forall x,y\in V\left(G\right)$ , where $\text{rad}\left(G\right)$ is the radius of the graph $G$ . The radial radio number of $\mathrm{\daleth }$ denoted by $rr\left(\mathrm{\daleth }\right)$ is the maximum number mapped under $\mathrm{\daleth }$ . The radial radio number of $G$ denoted by $rr\left(G\right)$ is equal to min { $rr\left(\mathrm{\daleth }\right)/\mathrm{\daleth }$ is a radial radio labelling of $G$ }.