Isomorphic

Definition

Isomorphic refers to two graphs that can be transformed into each other by renaming vertices without changing the structure or connections. Essentially, they have an identical form or shape in terms of vertex connections and edges.

5 Must Know Facts For Your Next Test

1. Two isomorphic graphs have the same number of vertices and edges.
2. Vertex degrees must be preserved between isomorphic graphs.
3. If one graph has a unique property like a specific cycle length, its isomorphic counterpart must also exhibit this property.
4. Graph invariants such as the degree sequence, number of cycles, and connectivity remain unchanged under isomorphism.
5. Determining if two large graphs are isomorphic can be computationally challenging and often requires algorithmic approaches.

Review Questions

• What properties must be preserved for two graphs to be considered isomorphic?
• Can two graphs with different degree sequences be isomorphic? Why or why not?
• Why might it be difficult to confirm whether two complex graphs are isomorphic?

Related terms

Subgraph: A subset of a graph's vertices and edges that forms a graph itself.

Homomorphism: A mapping between two algebraic structures that preserves their operations.

Graph Invariants: Properties of graphs that remain unchanged under graph isomorphisms, such as vertex degree sequence or the number of connected components.