Further results on the radio number of trees

TL;DR

This paper investigates the radio number of specific tree families, providing new exact values by analyzing graph operations on trees, which advances understanding of radio labelings in graph theory.

Contribution

It determines the radio number for three families of trees formed through graph operations, extending previous results in radio labeling of trees.

Findings

Exact radio numbers for three new tree families

Extension of radio labeling results to graph-constructed trees

Methodology for calculating radio numbers in complex trees

Abstract

Let $G$ be a finite, connected, undirected graph with diameter $diam(G)$ and $d(u,v)$ denote the distance between $u$ and $v$ in $G$. A radio labeling of a graph $G$ is a mapping $f: V(G) \rightarrow \{0,1,2,...\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u, v$ of $G$. The radio number of $G$, denoted by $rn(G)$, is the smallest integer $k$ such that $G$ has a radio labeling $f$ with $\max\{f(v) : v \in V(G)\} = k$. In this paper, we determine the radio number for three families of trees obtained by taking graph operation on a given tree or a family of trees.