Further results on the radio number of 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.