Let Γ1,Γ2 be general graphs.
A graph homomorphism𝑓:Γ1→Γ2 is a function V(𝑓):V(Γ1)→V(Γ2) which “almost preserves” the adjacency matrix, graph i.e.
|Γ1(𝑣,𝑤)|≤|Γ2(𝑓(𝑣),𝑓(𝑤))|
where if the inequality is made an equality 𝑓 is a full graph homomorphism.
The terms graph isomorphism, graph endomorphism, and graph automorphism are then defined accordingly,
and we have the Category of general graphs.