baselineskip=12pt Graph labeling assigns integers to vertices, edges, or both of a graph subject to prescribed balance conditions, and has broad applications in coding theory, network design, and communication protocols. In this paper we study the absolutely k-harmonious labeling (ABH labeling) of product graphs. A vertex labeling assigns to every vertex an integer label drawn from the residues modulo k, and every edge is then given an induced label formed from the absolute difference between the sum of its two endpoint labels and k, reduced modulo k. A labeling is called absolutely k-harmonious if, across all label classes, the vertex counts carrying each label differ from one another by at most one, and the edge counts carrying each label likewise differ by at most one. We prove that the absolutely k-harmonious property is preserved under three principal graph product operations ? the Cartesian product, the tensor product, and the strong product ? of two absolutely k-harmonious graphs, provided suitable divisibility conditions hold. We further prove that the property is preserved under graph complementation, when the order and the size of the graph are each divisible by k, and under disjoint union. Concretely, we establish that grid graphs formed as Cartesian products of two paths, and complete bipartite graphs whose part sizes sum to a multiple of three, are absolutely 3-harmonious for all suitable orders. All theorems are derived rigorously from the defining balance conditions and each is accompanied by a fully worked numerical example. The restriction to the case k equal to 3 for concrete graph families is natural because the modulo-3 cyclic structure aligns with the ternary residue classes of path indices, and extensions to larger values of k are identified as directions for future research.
| Published in | American Journal of Applied Mathematics (Volume 14, Issue 5) |
| DOI | 10.11648/j.ajam.20261405.14 |
| Page(s) | 303-306 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2026. Published by Science Publishing Group |
Absolutely k-harmonious Labeling, Product Graphs, Cartesian Product, Tensor Product, Strong Product, Complete Bipartite Graph, Balanced Labeling, ABH Labeling
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APA Style
Annathurai, K., Gomathi, M. (2026). Absolutely k-Harmonious Labeling of Product Graphs. American Journal of Applied Mathematics, 14(5), 303-306. https://doi.org/10.11648/j.ajam.20261405.14
ACS Style
Annathurai, K.; Gomathi, M. Absolutely k-Harmonious Labeling of Product Graphs. Am. J. Appl. Math. 2026, 14(5), 303-306. doi: 10.11648/j.ajam.20261405.14
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Download@article{10.11648/j.ajam.20261405.14,
author = {Kailasam Annathurai and Murugan Gomathi},
title = {Absolutely k-Harmonious Labeling of Product Graphs},
journal = {American Journal of Applied Mathematics},
volume = {14},
number = {5},
pages = {303-306},
doi = {10.11648/j.ajam.20261405.14},
url = {https://doi.org/10.11648/j.ajam.20261405.14},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20261405.14},
abstract = {baselineskip=12pt Graph labeling assigns integers to vertices, edges, or both of a graph subject to prescribed balance conditions, and has broad applications in coding theory, network design, and communication protocols. In this paper we study the absolutely k-harmonious labeling (ABH labeling) of product graphs. A vertex labeling assigns to every vertex an integer label drawn from the residues modulo k, and every edge is then given an induced label formed from the absolute difference between the sum of its two endpoint labels and k, reduced modulo k. A labeling is called absolutely k-harmonious if, across all label classes, the vertex counts carrying each label differ from one another by at most one, and the edge counts carrying each label likewise differ by at most one. We prove that the absolutely k-harmonious property is preserved under three principal graph product operations ? the Cartesian product, the tensor product, and the strong product ? of two absolutely k-harmonious graphs, provided suitable divisibility conditions hold. We further prove that the property is preserved under graph complementation, when the order and the size of the graph are each divisible by k, and under disjoint union. Concretely, we establish that grid graphs formed as Cartesian products of two paths, and complete bipartite graphs whose part sizes sum to a multiple of three, are absolutely 3-harmonious for all suitable orders. All theorems are derived rigorously from the defining balance conditions and each is accompanied by a fully worked numerical example. The restriction to the case k equal to 3 for concrete graph families is natural because the modulo-3 cyclic structure aligns with the ternary residue classes of path indices, and extensions to larger values of k are identified as directions for future research.},
year = {2026}
}
TY - JOUR T1 - Absolutely k-Harmonious Labeling of Product Graphs AU - Kailasam Annathurai AU - Murugan Gomathi Y1 - 2026/09/22 PY - 2026 N1 - https://doi.org/10.11648/j.ajam.20261405.14 DO - 10.11648/j.ajam.20261405.14 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 303 EP - 306 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20261405.14 AB - baselineskip=12pt Graph labeling assigns integers to vertices, edges, or both of a graph subject to prescribed balance conditions, and has broad applications in coding theory, network design, and communication protocols. In this paper we study the absolutely k-harmonious labeling (ABH labeling) of product graphs. A vertex labeling assigns to every vertex an integer label drawn from the residues modulo k, and every edge is then given an induced label formed from the absolute difference between the sum of its two endpoint labels and k, reduced modulo k. A labeling is called absolutely k-harmonious if, across all label classes, the vertex counts carrying each label differ from one another by at most one, and the edge counts carrying each label likewise differ by at most one. We prove that the absolutely k-harmonious property is preserved under three principal graph product operations ? the Cartesian product, the tensor product, and the strong product ? of two absolutely k-harmonious graphs, provided suitable divisibility conditions hold. We further prove that the property is preserved under graph complementation, when the order and the size of the graph are each divisible by k, and under disjoint union. Concretely, we establish that grid graphs formed as Cartesian products of two paths, and complete bipartite graphs whose part sizes sum to a multiple of three, are absolutely 3-harmonious for all suitable orders. All theorems are derived rigorously from the defining balance conditions and each is accompanied by a fully worked numerical example. The restriction to the case k equal to 3 for concrete graph families is natural because the modulo-3 cyclic structure aligns with the ternary residue classes of path indices, and extensions to larger values of k are identified as directions for future research. VL - 14 IS - 5 ER -
Department of Mathematics, Thiruvalluvar College, Papanasam, India
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, India
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