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 $\RR$ on $V$.

Digraph
A digraph $D$ is a non-empty set $V$ together with an antireflexive relation $\RR$ on $V$.

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

Loop-Digraph
A loop-digraph $D$ is a non-empty set $V$ together with a relation $\RR$ 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 Equivalence Relation.