MAKASSAR, Indonesia Feb 2 2026/graphtheory/ Imagine a network of roads, computers, or social connections in which every point must be recognized without confusion. Graph theory offers a mathematical way to study this problem. In a graph, points are called vertices and the links between them are called edges. The article “Another Antimagic Conjecture” explores how numbers can be assigned to vertices so that every point receives a unique total value based on its surroundings.
The idea is called D-antimagic labeling. First, every vertex is given a different number. Then, for each vertex, researchers add the labels of other vertices located at selected distances, represented by a set . The resulting sum becomes that vertex’s weight. A labeling is D-antimagic when all vertices have different weights.
This can be compared to giving every house in a city a unique identity based not only on its own number, but also on the numbers of nearby houses. If two houses have exactly the same group of neighbors at the chosen distances, they will always produce the same total, no matter how the labels are arranged. The article calls such vertices D-twins.
The researchers propose a bold conjecture: a graph has a D-antimagic labeling if and only if it has no pair of D-twins. In other words, the only obstacle should be two vertices that share exactly the same D-neighborhood. The study does not claim a complete proof for every graph. Instead, it presents several strong pieces of evidence.
One major contribution is computational. For D={1}, meaning that only directly connected neighbors are counted, the researchers tested all non-isomorphic graphs with up to eight vertices. Every graph without twin neighborhoods admitted a suitable antimagic labeling. For graphs with eight vertices, 12,346 different non-isomorphic structures were examined, and all 8,047 graphs without twins matched the conjecture.
The paper also proves that certain D-antimagic graphs remain antimagic when combined as separate components. This applies to several families, including cycles, complete graphs, hypercubes, paths, wheels, fans, friendship graphs, and sun graphs under stated conditions. The diagrams in the article show how carefully arranged labels create distinct weights even across repeated or disconnected structures.
These findings matter because unique identification is important in communication networks, coding, scheduling, sensor systems, and data organization. The work provides both theoretical ideas and computational methods that future researchers can develop further.
This research directly supports Sustainable Development Goal 4: Quality Education. It expands advanced mathematical knowledge, demonstrates the value of computational experimentation, and provides accessible examples for teaching graph theory, algorithms, and logical reasoning. It also shows students that an unproven conjecture can still guide meaningful research.
Ultimately, the article reveals a simple but powerful principle: when two points do not see exactly the same neighborhood, mathematics may be able to give each of them a unique identity.
Reference:
DOI:Â https://doi.org/10.3390/sym13112071
Contact:
Prof. Nurdin
+62 813-9519-0801
nurdin1701@unhas.ac.id
