A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1, 2,\ldots\}$ such that $|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$ for every pair of distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$ of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$ with $\max\{f(v):v \in V(G)\} = k$. The weight of a tree $T$ from a vertex $v \in V(T)$ 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 center of $T$. It is known that any tree has one or two weight centers. A tree is called a two-branch tree if the removal of all its weight centers 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.

翻译：图$G$的无线电标号是一个映射$f : V(G) \rightarrow \{0, 1, 2,\ldots\}$，满足对于$G$中任意两个不同顶点$u, v$，有$|f(u)-f(v)| \geq diam(G) + 1 - d(u,v)$，其中$diam(G)$是$G$的直径，$d(u,v)$是$G$中$u$与$v$之间的距离。图$G$的无线电数$rn(G)$是使得$G$存在一个无线电标号$f$且$\max\{f(v):v \in V(G)\} = k$的最小整数$k$。树$T$中从顶点$v \in V(T)$出发的权重是指$T$中从$v$到所有其他顶点的距离之和，而$T$中达到最小权重的顶点称为$T$的权重中心。已知任意树都有一个或两个权重中心。若删除树的所有权重中心后得到的森林恰好有两个连通分支，则称该树为双分支树。本文得到了双分支树无线电数的一个紧下界，该下界改进了已知的一般树的下界。我们还给出了达到该改进下界的充要条件。利用这些结果，我们确定了两类逐层正则双分支树的无线电数。