Graph Isomorphism is Equivalence Relation

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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 Equivalence Relation.

$\blacksquare$


Sources