MAKASSAR, Indonesia June 15 2026/graphtheory/ Imagine a city map made only of dots and connecting roads. In mathematics, this structure is called a graph: the dots are vertices, while the connecting lines are edges. Graphs can represent transportation routes, communication systems, social relationships, computer networks, and many other connected structures. A recent study explores a fascinating question: how can numbers be assigned to the edges so that every vertex receives a unique mathematical identity?
The study, titled “Modular Irregularity Strength of Vertex Amalgamation and Comb Product Path with Cycle Related Graphs,” investigates a special technique known as modular irregular labeling. In simple terms, each edge is given a number. The numbers on the edges connected to a vertex are then added and interpreted using modular arithmetic, which works much like the repeating numbers on a clock. The goal is to make the final value at every vertex different.
This may sound like a mathematical puzzle, but it addresses an important issue in network theory: how to distinguish individual points in a connected system using a limited set of labels. The smallest maximum label needed to create these unique values is called the modular irregularity strength.
The researchers focused on several complex graph families. One of them is the Dutch windmill graph, formed by joining several square-shaped cycles at one common vertex. They determined the exact modular irregularity strength for this structure under specific numerical conditions. The study also examined “comb product” graphs, which resemble a main path or cycle with additional cycles attached along it, much like teeth connected to a comb.
A major finding is that several different comb-product structures share the same general formula for their modular irregularity strength. These include combinations of paths, cycles, regular graphs, and circulant graphs. However, the researchers also identified conditions where modular irregular labeling is impossible. In particular, certain graphs whose number of vertices leaves a remainder of two when divided by four have no valid modular irregular labeling.
The value of this research lies in its ability to reveal order within complicated networks. By proving exact formulas, the authors provide future researchers with reliable tools instead of requiring them to test labels one by one. Such mathematical foundations can support further studies in network identification, coding, scheduling, and structural optimization.
This work is directly connected to Sustainable Development Goal 4: Quality Education. Advanced mathematical research strengthens scientific knowledge, supports high-quality university teaching, and provides new material for students studying graph theory, discrete mathematics, and network science. By presenting difficult mathematical ideas in understandable ways, research like this can also encourage broader public interest in mathematics.
Ultimately, the study shows that even abstract numbers placed on simple lines can create unique identities, predictable patterns, and deeper understanding of complex connected systems.
Reference:
DOI: https://dx.doi.org/10.5614/ejgta.2026.14.1.4
Contact:
Prof. Nurdin
+62 813-9519-0801
nurdin1701@unhas.ac.id
