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Copyright 2026 The Formal Conjectures Authors.
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you may not use this file except in compliance with the License.
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-/
module
public import Mathlib.Algebra.Polynomial.Roots
public import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
public import Mathlib.Data.Real.Basic
public import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic@[expose] public sectionnamespace SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α]
Given a simple graph G with adjacency matrix A, this is the number n₀ + min n₊ n₋ where:
n₀ is the multiplicity of zero as an eigenvalue of A
n₊ is the number of positive eigenvalues of A (counting multiplicities)
n₋ is the number of negative eigenvalues of A (counting multiplicities)
noncomputable def cvetkovic (G : SimpleGraph α) [DecidableRel G.Adj] : ℕ :=
letI A : Matrix α α ℝ := G.adjMatrix ℝ
letI spectrum : Multiset ℝ := (Matrix.charpoly A).roots
letI positive_count : ℕ := spectrum.countP (fun x => 0 < x)
letI negative_count : ℕ := spectrum.countP (fun x => 0 > x)
letI zero_count : ℕ := spectrum.countP (fun x => 0 = x)
zero_count + min positive_count negative_countend SimpleGraph