Some techniques to find large lower bound trees for the radio number

For a simple finite connected graph $G$, let $diam(G)$ and $d_G(u,v)$ denote the diameter of $G$ and distance between $u$ and $v$ in $G$, respectively. A radio labeling of a graph $G$ is a mapping $f$ : $V(G)\to \{0,1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$ such that $\mathrm{\mid }f(u)-f(v)\mathrm{\mid }\ge diam(G)+1-d_G(u,v)$ holds for every pair of distinct vertices $u,v$ of $G$. The radio number $rn(G)$ of $G$ is the smallest number $k$ such that $G$ has radio labeling $f$ with max$\{f(v):v\in V(G)\}$ = $k$. Bantva et al. gave a lower bound for the radio number of trees in [Lemma 3.1, Discrete Applied Math.,217(2017),110-122] and, a necessary and sufficient condition to achieve this lower bound in [Theorem 3.2, Discrete Applied Math.,217(2017),110-122]. Denote the lower bound for the radio number of trees given in [Lemma 3.1, Discrete Applied Math.,217(2017),110-122] by $lb(T)$. A tree $T$ is called a lower bound tree for the radio number if $rn(T)$ = $lb(T)$. In this paper, we construct some large lower bound trees for the radio number using known lower bound trees.