On Distance-Based Arithmetic Radio Number of Standard Graph Classes

Dr. Ramya Hebbar; Sooryanarayana  B; Vishukumar  M; Sneha  G. Kulkarni

doi:10.5269/bspm.80132

Dr. Ramya Hebbar Department of Mathematics, Nitte Meenakshi Institute of Technology, NITTE (Deemed to be ) University, Bengaluru https://orcid.org/0009-0004-3950-2881
Sooryanarayana B Nitte Meenakshi Institute of Technology, NITTE (Deemed to be) University, Bengaluru https://orcid.org/0000-0002-2835-2855
Vishukumar M Department of Mathematics, School of Applied Sciences, REVA University, Bengaluru. https://orcid.org/0000-0003-0719-3637
Sneha G. Kulkarni Department of Mathematics, Dr. Ambekar Institute of Technology, Bengaluru. https://orcid.org/0009-0007-4949-280X

Abstract

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.

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