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doc: rename Zhou → Zhou-3, re-add visualization links
All references to "Zhou graph" now say "Zhou-3 graph" to distinguish from the 6-regular Zhou-6 quotient. Re-adds image links that were lost during the Zhou-6 edge list addition. Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>
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Archive/ZhouGraph.lean

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@@ -7,9 +7,9 @@ import Mathlib.Combinatorics.SimpleGraph.Basic
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import Mathlib.Combinatorics.SimpleGraph.QuotientGraph
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/-!
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# The Zhou graph (F182A) and its Z₂ quotient
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# The Zhou-3 graph (F182A) and its Z₂ quotient (Zhou-6)
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The Zhou graph (Foster census F182A) is a cubic arc-transitive graph on 182 vertices.
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The **Zhou-3 graph** (Foster census F182A) is a cubic arc-transitive graph on 182 vertices.
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- **Sabidussi**: Sab(PSL(2,13), S₃), |PSL(2,13)| = 1092
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- **Imprimitive**: S₃ is NOT maximal — sits inside D₁₂ (dihedral order 12, index 91)
@@ -18,9 +18,14 @@ The Zhou graph (Foster census F182A) is a cubic arc-transitive graph on 182 vert
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The **Zhou-6 graph** is the Z₂ quotient: Sab(PSL(2,13), D₁₂), a 6-regular graph on
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91 vertices. D₁₂ IS maximal in PSL(2,13), so the quotient is **primitive**.
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## Visualizations
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* [The Zhou-3 graph](https://raw.githubusercontent.com/RaggedR/symmetric-graphs/main/lean/named_graphs/zhou3-F182A.jpg) — 182 vertices, imprimitive block structure
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* [The Zhou-6 quotient](https://raw.githubusercontent.com/RaggedR/symmetric-graphs/main/lean/named_graphs/zhou6-91v.jpg) — 91-vertex quotient by the Z₂ block system
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## Main results
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* `zhouGraph_regular` — the Zhou graph is 3-regular
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* `zhouGraph_regular` — the Zhou-3 graph is 3-regular
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* `zhou6Graph_regular` — the Zhou-6 graph is 6-regular
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* `zhou6_eq_quotient` — the Zhou-6 graph equals `zhouGraph.quotientGraph zhouBlockMap`
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@@ -31,7 +36,7 @@ The **Zhou-6 graph** is the Z₂ quotient: Sab(PSL(2,13), D₁₂), a 6-regular
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set_option linter.style.nativeDecide false
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/-! ### The Zhou graph (3-regular, 182 vertices) -/
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/-! ### The Zhou-3 graph (3-regular, 182 vertices) -/
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private def zhouEdges : List (Fin 182 × Fin 182) := [
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(0,3),(0,37),(0,68),(1,4),(1,38),(1,71),
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private def zhouAdjBool (u v : Fin 182) : Bool :=
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zhouEdges.any fun (a, b) => (u == a && v == b) || (u == b && v == a)
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/-- The **Zhou graph** (F182A): Sab(PSL(2,13), S₃), cubic, 182 vertices. Imprimitive. -/
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/-- The **Zhou-3 graph** (F182A): Sab(PSL(2,13), S₃), cubic, 182 vertices. Imprimitive. -/
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def zhouGraph : SimpleGraph (Fin 182) where
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Adj u v := zhouAdjBool u v
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symm u v := by simp only [zhouAdjBool]; revert u v; native_decide
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instance : DecidableRel zhouGraph.Adj :=
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fun u v => inferInstanceAs (Decidable (zhouAdjBool u v))
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/-- The Zhou graph is 3-regular. -/
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/-- The Zhou-3 graph is 3-regular. -/
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theorem zhouGraph_regular :
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∀ v : Fin 182, (Finset.univ.filter fun w => zhouGraph.Adj v w).card = 3 := by
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native_decide
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/-- The Zhou graph has 273 edges. -/
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/-- The Zhou-3 graph has 273 edges. -/
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theorem zhouGraph_edgeCount :
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(Finset.univ.filter fun p : Fin 182 × Fin 182 =>
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p.1 < p.2 ∧ zhouGraph.Adj p.1 p.2).card = 273 := by
@@ -120,7 +125,7 @@ private def zhouBlockData : List (Fin 91) := [
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private theorem zhouBlockData_length : zhouBlockData.length = 182 := by native_decide
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/-- The Z₂ block map: `Fin 182 → Fin 91`. -/
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/-- The Z₂ block map on the Zhou-3 graph: `Fin 182 → Fin 91`. -/
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def zhouBlockMap (v : Fin 182) : Fin 91 :=
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zhouBlockData.get (v.cast zhouBlockData_length.symm)
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@@ -198,12 +203,12 @@ theorem zhou6Graph_edgeCount :
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/-! ### Quotient relationship
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The Zhou-6 graph is the Z₂ quotient of the Zhou graph via `zhouBlockMap`.
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The Zhou-6 graph is the Z₂ quotient of the Zhou-3 graph via `zhouBlockMap`.
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The brute-force `native_decide` proof of `zhou6_eq_quotient` is too expensive
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(existential over Fin 182² for each of 91² pairs). A structural proof via
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PSL(2,13) generators (analogous to `G2Action.langer_eq_tutte12_distance2'`)
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would scale better. -/
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/-- The Z₂ quotient of the Zhou graph (defined abstractly via quotientGraph). -/
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/-- The Z₂ quotient of the Zhou-3 graph (defined abstractly via quotientGraph). -/
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def zhouQuotientGraph : SimpleGraph (Fin 91) :=
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zhouGraph.quotientGraph zhouBlockMap

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