Graph Isomorphism is Equivalence Relation

Theorem
Graph isomorphism is an equivalence relation.

Proof
From the formal definitions:

Simple Graph
A (simple) graph $$G$$ is a non-empty set $$V$$ together with an antireflexive, symmetric relation $$E$$ on $$V$$.

Digraph

 * A digraph $$D$$ is a non-empty set $$V$$ together with an antireflexive relation $$E$$ on $$V$$.

Loop-graph
A loop-graph $$P$$ is a non-empty set $$V$$ together with a symmetric relation $$E$$ on $$V$$.

Loop-Digraph
A loop-digraph $$D$$ is a non-empty set $$V$$ together with a relation $$E$$ on $$V$$.

It can be seen from these definitions that all the above are relational structures with more or less restriction on the relation.

Hence the result from Relation Isomorphism is an Equivalence.