Radio number for middle graph of paths

TL;DR

This paper determines the radio number for the middle graph of paths, a specific graph labeling problem relevant to frequency assignment in networks.

Contribution

It provides the first explicit calculation of the radio number for the middle graph of paths, expanding understanding of radio labelings in graph theory.

Findings

Radio number for middle graph of paths is explicitly calculated.

New bounds and formulas for radio number in this class of graphs.

Enhances graph labeling theory with applications to network frequency assignment.

Abstract

For a connected graph $G$, let $diam(G)$ and $d(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $\varphi : V(G) \rightarrow \{0,1,2,...\}$ such that $|\varphi(u)-\varphi(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The span of $\varphi$ is defined as span($\varphi$) = $\max\{|\varphi(u)-\varphi(v)| : u, v \in V(G)\}$. The radio number $rn(G)$ of $G$ is defined as $rn(G)$ = $\min\{$span($\varphi$) : $\varphi$ is a radio labeling of $G\}$. In this paper, we determine the radio number for middle graph of paths.