Graph Isomorphism in Discrete Mathematics: Determining Structural Equivalence of Graphs

Understand graph isomorphism, a concept in discrete mathematics that determines whether two graphs have the same structure, regardless of their visual representation. This guide defines graph isomorphism, outlines the necessary conditions for isomorphism, and provides examples to illustrate this key concept.

Graph Isomorphism in Discrete Mathematics

What is Graph Isomorphism?

Two graphs are isomorphic if they have the same structure, even if they look different. Imagine you have two graphs; if you can rename the vertices of one graph and rearrange the edges in a way that it becomes identical to the other graph, then those graphs are isomorphic. It's like having two different drawings of the same underlying network.

Necessary Conditions for Graph Isomorphism

For two graphs to be isomorphic, they must satisfy these conditions:

  1. Equal Number of Vertices: Both graphs must have the same number of vertices.
  2. Equal Number of Edges: Both graphs must have the same number of edges.
  3. Equal Degree Sequence: The degree sequence (list of vertex degrees in ascending order) must be identical for both graphs.
  4. Identical Cycle Structure: If a graph has a cycle of length k, the other graph must also have a cycle of length k.

(The note that the degree sequence is the list of vertex degrees in ascending order should be included here.)

Important Points Regarding Isomorphism

  • These conditions are necessary but not sufficient. Meeting all these conditions doesn't guarantee isomorphism; further checks are often needed.
  • If any of these conditions is not met, the graphs are definitely not isomorphic.

Sufficient Conditions for Graph Isomorphism

These conditions, if met, guarantee that two graphs are isomorphic:

  • Their complement graphs are isomorphic.
  • Their adjacency matrices are identical.
  • The graphs resulting from deleting corresponding vertices are isomorphic.

Examples: Checking for Graph Isomorphism

Example 1: Non-Isomorphic Graphs

(An example with two graphs that are not isomorphic because they have different numbers of edges would be included here. The relevant conditions for isomorphism would be checked.)

Example 2: Checking Isomorphism Between Three Graphs

(Three graphs would be given here. Conditions 1, 2, 3, and 4 would be checked for all pairs of graphs. It would be demonstrated that two of the graphs satisfy all necessary conditions, and the sufficient condition of isomorphic complement graphs would be used to conclude isomorphism for those two graphs. The third graph would be shown to be non-isomorphic to the others.)

Example 3: Non-Isomorphic Graphs Due to Cycle Structure

(Two graphs would be given here. It would be demonstrated that while they satisfy the first three necessary conditions, they are not isomorphic because they have different cycle structures.)

Conclusion

Graph isomorphism is a fundamental concept in graph theory. Determining whether two graphs are isomorphic can be computationally challenging, but understanding the necessary and sufficient conditions is crucial for analyzing and comparing graph structures.