MAKASSAR, Indonesia May 15 2026/graphtheory/ Flowers are often admired for their symmetry, but in mathematics, flower-like patterns can also help researchers understand complex networks. A recent study investigates two special structures—the modified sunflower graph and the flower petal graph—and develops a mathematical method for giving every point in these networks a unique identity.
In graph theory, a network is represented by vertices and edges. Vertices may stand for devices, stations, users, or control points, while edges represent the connections between them. The researchers focused on total vertex irregular labeling, a method that assigns numbers to both vertices and edges. For each vertex, its own label is added to the labels of all connected edges. The resulting total is called the vertex weight.
The challenge is to make every vertex weight different while using the smallest possible range of numbers. This minimum range is known as the total vertex irregularity strength. In simple terms, it answers an efficiency question: how few labels are needed so that every point in a network can still be distinguished?
The study uses a constructive method. Instead of only proving that a solution exists, the authors provide explicit rules for assigning labels. They then calculate the resulting vertex weights and show that no two are the same. This makes the method systematic, repeatable, and adaptable to other networks with similar structures.
The first structure is the modified sunflower graph. It contains a repeating, symmetrical arrangement of vertices and connections. For an even integer of at least four, the researchers found that its total vertex irregularity strength is:
The second structure, called the flower petal graph, has longer petal-like branches surrounding a central cycle. Because this network is more complex, it requires a larger label range. Its total vertex irregularity strength is:
The article’s diagrams make these results easier to visualize. The modified sunflower example shows layered circular connections, while the flower petal example resembles a ring surrounded by repeating petals. Additional labeled diagrams demonstrate how the assignments produce different weights for every vertex.
Although these results belong to pure mathematics, the method has practical potential. Flower-shaped graphs resemble hub-and-spoke sensor systems, circular communication networks, and other repeated network designs. Unique vertex weights could help distinguish devices, allocate communication loads, reduce signal collisions, or organize bandwidth and energy use.
This research directly supports Sustainable Development Goal 4: Quality Education. It expands knowledge in graph theory, offers a method that students and researchers can reproduce, and demonstrates how abstract mathematics can contribute to network modeling. Presenting such ideas in accessible language also helps connect advanced university research with wider public understanding.
Ultimately, the study shows that symmetry does not have to create confusion. With carefully constructed labels, every point in a repetitive network can still have its own unmistakable mathematical identity.
Reference:
DOI: https://doi.org/10.1016/j.mex.2025.103587
Contact:
Prof. Nurdin
+62 813-9519-0801
nurdin1701@unhas.ac.id
