On Distance-Based Arithmetic Radio Number of Standard Graph Classes

Résumé

Let k∈Z+ and let G=(V,E) be a connected graph of order n. An arithmetic k-radiolabeling is a bijection η:V→{1,1+k,1+2k,...,1+(n−1)k} suchthat for any two distinct vertices u,v∈V,  the
 condition|η(u)−η(v)|>diam(G)−dist(u,v) is satisfied. The least such k is defined as the arithmetic radionumber of G, denoted by Ra(G). In this paper, we establish exact values of Ra(G) for several families of graphs, including paths, cycles, squares of paths, and the join of graphs. Our results contribute to the broader context of distance constrained labeling by combining structural graph properties with arithmetic progressions in labeling.