Graph homomorphism

Let $Γ 1, Γ 2$ be general graphs. A graph homomorphism $𝑓 : Γ 1 \to Γ 2$ is a function $V (𝑓) : V (Γ 1) \to V (Γ 2)$ which “almost preserves” the adjacency matrix, #m/def/graph i.e.

| Γ 1 (𝑣, 𝑤) | \leq | Γ 2 (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 $𝖦 𝗋 𝗉 𝗁$ .

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