Radio Number for Generalized Petersen Graphs <inline-formula> <tex-math notation="LaTeX">$P(n,2)$ </tex-math></inline-formula>

Abstract

Let $G$ be a connected graph and $d(\mu,\omega)$ be the distance between any two vertices of $G$ . The diameter of $G$ is denoted by $diam(G)$ and is equal to $\max \{d(\mu,\omega); \\ \mu,\omega \in G\}$ . The radio labeling (RL) for the graph $G$ is an injective function $\digamma:V(G)\rightarrow N\cup \{0\}$ such that for any pair of vertices $\mu $ and $\omega \,\,|\digamma (\mu)-\digamma (\omega)|\geq diam(G)-d(\mu,\omega)+1$ . The span of radio labeling is the largest number in $\digamma (V)$ . The radio number of $G$ , denoted by $rn(G)$ is the minimum span over all radio labeling of $G$ . In this paper, we determine radio number for the generalized Petersen graphs, $P(n,2)$ , $n=4k+2$ . Further the lower bound of radio number for $P(n,2)$ when $n=4k$ is determined.

Keywords

• Diameter
• radio number
• generalized Petersen graph