Radio labelling of two-branch trees

A radio labelling of a graph G is a mapping $f:V\left(G\right)\to \{0,1,2,\dots \}$ such that $\left|f\right(u)-f(v\left)\right|\ge \mathrm{diam}\left(G\right)+1-d(u,v)$ for every pair of distinct vertices $u,v$ of G, where $\mathrm{diam}\left(G\right)$ is the diameter of G and $d(u,v)$ is the distance between u and v in G. The radio number $\mathrm{rn}\left(G\right)$ of G is the smallest integer k such that G admits a radio labelling f with $\max \left\{f\right(v):v\in V(G\left)\right\}=k$. The weight of a tree T from a vertex $v\in V\left(T\right)$ is the sum of the distances in T from v to all other vertices, and a vertex of T achieving the minimum weight is called a weight centre of T. It is known that any tree has one or two weight centres. A tree is called a two-branch tree if the removal of all its weight centres results in a forest with exactly two components. In this paper we obtain a sharp lower bound for the radio number of two-branch trees which improves a known lower bound for general trees. We also give a necessary and sufficient condition for this improved lower bound to be achieved. Using these results, we determine the radio number of two families of level-wise regular two-branch trees.