Lax Lean Archive

Homomorphism counts

Lax871432.HomomorphismCounts · concepts/Lax871432/HomomorphismCounts.lean · lax-871432

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    Natural Language Statement

    Definition

    A homomorphism from a graph FF to a graph GG is a map V(F)→V(G)V(F) \to V(G) sending adjacent vertices to adjacent vertices. We write hom⁡(F,G)\hom(F, G) for the number of such maps.

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    • page 3 of this submission's paper

    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Maps
    2import Mathlib.SetTheory.Cardinal.Finite
    3
    4/-!
    5---
    6title: Homomorphism counts
    7type: definition
    8---
    9A *homomorphism* from a graph FF to a graph GG is a map V(F)→V(G)V(F) \to V(G) sending adjacent
    10vertices to adjacent vertices. We write hom⁡(F,G)\hom(F, G) for the number of such maps.
    11-/
    12
    13namespace Lax871432.HomomorphismCounts
    14
    15/-- hom⁡(F,G)\hom(F, G), the number of homomorphisms from `F` to `G`. -/
    16noncomputable def homCount {V W : Type*} (F : SimpleGraph V) (G : SimpleGraph W) : ℕ :=
    17 Nat.card (F →g G)
    18
    19end Lax871432.HomomorphismCounts
    20
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    From Mathlib

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