Graph Isomorphism is an Equivalence

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.

$\blacksquare$

Sources

• Gary Chartrand: Introductory Graph Theory (1977): $\S 2.2$: Theorem $2.3$