/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Defs
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Combinatorics.SimpleGraph.Basic@[expose] public sectionnamespace SimpleGraphvariable {α : Type*} [Fintype α] [DecidableEq α]Fractional alpha. This is defined as $$ \max_x \sum_{v \in V} x_v $$ subject to $0 \le x_v \le 1$ for all $v \in V$ and $x_u + x_v \le 1$ whenever $(u, v)$ are connected by an edge.
noncomputable def fractionalAlpha (G : SimpleGraph α) [DecidableRel G.Adj] : ℝ :=
sSup {(∑ i, x i) | (x : α → ℝ) (_ : ∀ v, x v ∈ Set.Icc 0 1)
(_ : ∀ u v, G.Adj u v → x u + x v ≤ 1)}end SimpleGraph