Giving Every Point a Unique Identity: The Mathematics Behind Symmetrical Networks

MAKASSAR, Indonesia Apr 2 2026/graphtheory/ In mathematics, this structure is called a dodecahedral graph. It contains points connected by lines in a highly regular pattern. Because of its symmetry, many parts of the network can look identical. This creates an interesting challenge: how can every point be given a unique mathematical identity without using unnecessarily large numbers?

A study titled “Two Types Irregular Labelling on Dodecahedral Modified Generalization Graph” explores this question through graph labelling. In graph theory, points are called vertices and connecting lines are called edges. Labels are numbers assigned to these elements. The goal of irregular labelling is to create a different weight for every vertex, even when the surrounding structure appears repetitive.

The researchers began with the familiar dodecahedral graph, which has 20 vertices and 30 edges. They modified its structure by adding connections and then expanded it into a wider family of graphs called the dodecahedral modified generalization graph, written as $GD_n$GDn​. The illustration in Figure 1 of the article shows the progression from the original dodecahedral graph to the modified and generalized versions.

The study examines two labelling systems. The first is total vertex irregular labelling. Here, numbers are assigned to both vertices and edges. The weight of a vertex is calculated by adding its own label to the labels of all edges connected to it. The researchers determined that, for even $n$n of at least six, the smallest required maximum label is:$$tvs(GD_n)=\left\lceil\frac{2n+3}{6}\right\rceil.$$

The second system is modular irregular labelling. In this approach, only the edges receive labels, and each vertex weight is calculated using modular arithmetic. This is similar to a clock, where numbers repeat after reaching a fixed limit. For even $n$n of at least six, the result is:$$ms(GD_n)=\left\lceil\frac{2n+2}{5}\right\rceil.$$

However, when $n$n is odd, modular irregular labelling is impossible for this graph family. The study expresses this result as $ms(GD_n)=\infty$

These formulas are valuable because they replace trial-and-error calculations with exact rules. Researchers studying other graphs with similar structures may adapt the same labelling patterns. The ideas could also contribute to network identification, coding, communication design, and systems in which many components must remain distinguishable.

This research is directly connected to Sustainable Development Goal 4: Quality Education. It enriches mathematical knowledge, provides clear theorems for teaching graph theory, and offers examples of how abstract reasoning can solve structural problems. Making such ideas understandable to wider audiences can also help students see mathematics not as a collection of formulas, but as a language for organizing complex systems.

Ultimately, the study shows that even a highly symmetrical network can give every point a unique identity when numbers are assigned with precision.

Reference:

DOI: https://doi.org/10.1016/j.heliyon.2022.e11197

Contact:
Prof. Nurdin
+62 813-9519-0801
nurdin1701@unhas.ac.id