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Binary cut-rank

Lax183560.CutRank · concepts/Lax183560/CutRank.lean · lax-183560

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    Natural Language Statement

    Definition

    For a vertex set AA in a graph, the cut-rank is the rank over the two-element field of the adjacency matrix with rows indexed by AA and columns indexed by the vertices outside AA.

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    Lean source view on GitHub

    1import Mathlib.Combinatorics.SimpleGraph.Finite
    2import Mathlib.LinearAlgebra.Matrix.Rank
    3import Mathlib.Data.ZMod.Basic
    4
    5/-!
    6---
    7title: Binary cut-rank
    8type: definition
    9---
    10For a vertex set AA in a graph, the cut-rank is the rank over the
    11two-element field of the adjacency matrix with rows indexed by AA and
    12columns indexed by the vertices outside AA.
    13-/
    14
    15namespace Lax183560.CutRank
    16
    17/-- Binary rank of the adjacency matrix across a cut. -/
    18noncomputable def cutRank {V : Type} [Fintype V] [DecidableEq V]
    19 (G : SimpleGraph V) (A : Finset V) : ℕ := by
    20 classical
    21 let M : Matrix {v : V // v ∈ A} {v : V // v ∉ A} (ZMod 2) :=
    22 fun u v => if G.Adj u v then 1 else 0
    23 exact M.rank
    24
    25
    26end Lax183560.CutRank
    27
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    From Mathlib

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