Antipodal number of full m-ary trees

Abstract

Let G=(V(G),E(G)) be a simple connected graph. For any two distinct vertices u and v, d(u,v) represents the distance between u and v. Suppose k be any positive integer with 1⩽k⩽diam(G), where the diam(G) represents the diameter of G. A radio k-labeling of G is a mapping f:V(G)→{0,1,2,…} such that |f(u)−f(v)|⩾k+1−d(u,v) for each pair of distinct vertices u and v of G. The absolute difference of the largest and smallest values in f(V(G)) is termed as the span of f, and is denoted by span(f). The antipodal number is the minimum span of a radio (diam(G)−1)-labeling of G and the radio number is the minimum span of a radio diam(G)-labeling of G. In this article we determine the antipodal number of the m-ary tree for any m≥3 with any height h≥3 and construct explicitly an optimal antipodal labeling.