Let G be a simple connected graph. For any two vertices u and v, let d(u,v) denote the distance between u and v in G, and let diam(G) denote the diameter of G. A radio-labeling of G is a function f which assigns to each vertex a non-negative integer (label) such that for every distinct vertices u and v in G, it holds that |f(u)−f(v)|≥diam(G)−d(u,v)+1. The span of f is the difference between the largest and smallest labels of f(V). The radio number of G, denoted by rn(G), is the smallest span of a radio labeling admitted by G. In this paper, we give a lower bound for the radio number of the Cartesian product of two trees. Moreover, we present three necessary and sufficient conditions, and three sufficient conditions for the product of two trees to achieve this bound. Applying these results, we determine the radio number of the Cartesian product of two stars as well as a path and a star.

A radio labeling of a graph G is a mapping f : V ( G ) → {0, 1, 2,...} such that | f ( u ) − f ( v ) | ⩾ d ( G ) + 1 − d ( u , v ) holds for every pair of vertices u and v , where d ( G ) is the diameter of G and d ( u , v ) is the distance between u and v in G . The radio number of G , denoted by r n ( G ) , is the smallest t such that G admits a radio labeling with t = max ⁡ { | f ( v ) − f ( u ) | : v , u ∈ V ( G ) } . A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is established. The block graph which achieves this bound is called a lower bound block graph . We prove three necessary and sufficient conditions for lower bound block graphs. Moreover, we give three sufficient conditions for a graph to be a lower bound block graph. Using these results, we present several families of lower bound block graphs, including the level-wise regular block graphs and the extended star of blocks. The line graph of a graph G ( V , E ) has E ( G ) as the vertex set, where two vertices are adjacent if they are incident edges in G . We extend our results to trees as trees and its line graphs are block graphs. We prove that if a tree is a lower bound block graph then, under certain conditions, its line graph is also a lower bound block graph, and vice versa. Consequently, we show that the line graphs of many known lower bound trees, excluding paths, are lower bound block graphs.