Definition:Isomorphism (Graph Theory)

Definition
Let $$G = \left({V \left({G}\right), E \left({G}\right)}\right)$$ and $$H = \left({V \left({H}\right), E \left({H}\right)}\right)$$ be graphs.

Let there exist a bijection $$F: V \left({G}\right) \to V \left({H}\right)$$ such that for each edge $$\left\{{u, v}\right\} \in E \left({G}\right)$$, there is an edge $$\left\{{F \left({u}\right), F \left({v}\right)}\right\} \in E \left({H}\right)$$.

That is, that:
 * $$F: V \left({G}\right) \to V \left({H}\right)$$ is a homomorphism, and


 * $$F^{-1}: V \left({H}\right) \to V \left({G}\right)$$ is a homomorphism.

Then $$G$$ and $$H$$ are isomorphic, and this is denoted $$G \cong H$$.

The function $$F$$ is called an isomorphism from $$G$$ to $$H$$.

It follows from this definition that Graph Isomorphism is an Equivalence.