A cubic graph, also known as a 3-regular graph, is a type of graph in which every vertex has exactly three edges connected to it.
This specific uniformity gives cubic graphs a range of interesting mathematical properties and applications in various areas such as network design, chemistry, and physics.
Properties of Cubic Graphs
1. Edge Count:
For a cubic graph with 𝑛n vertices, the total number of edges 𝐸E is 3𝑛223n.
This follows because each vertex contributes three edges, but this counts each edge twice (once at each end).
2. Hamiltonian Graphs:
Many cubic graphs are Hamiltonian, meaning there is a closed path that visits each vertex exactly once.
However, not all cubic graphs have this property.
3. Eulerian Path and Cycle:
Since every vertex has an even degree (three), every connected cubic graph has an Eulerian circuit (a path that uses each edge exactly once and returns to the starting vertex).
4. Color ability:
Every cubic graph without bridges (edges whose removal increases the number of connected components) is 3-edge-colorable.
This means you can color the edges with three colors such that no two adjacent edges share the same color.
5. Symmetry:
Cubic graphs often exhibit a high degree of symmetry, making them a subject of interest in graph theory and combinatorial design.
6. Girth:
The girth of a graph is the length of its shortest cycle.
For cubic graphs, the girth can vary, but for certain well-known cubic graphs like the Petersen graph, the girth is relatively small.
Example of simple cubic graph
The diagram above shows an example of a cubic graph. In this graph:
Each vertex is connected to exactly three other vertices, demonstrating the defining property of cubic graphs.
The layout and structure highlight the symmetry and regularity that are typical in cubic graphs, making them interesting both visually and mathematically.