feat(Combinatorics/SimpleGraph/Acyclic): define star graphs

[8e7]

Add a new definition starGraph and several key lemmas. Star graphs are a trivial class of tree, often used in constructive proofs regarding trees.

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All lemmas are hand written first, then golfed with the help of Claude Code.

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PR summary 2ff88851d5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ connected_starGraph
+ instance [DecidableEq V] (r : V) : DecidableRel (starGraph r).Adj := by
+ isAcyclic_starGraph
+ isTree_starGraph
+ starGraph
+ starGraph_adj
+ starGraph_center_adj
+ starGraph_center_degree
+ starGraph_not_center_imp_degree_one

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[vlad902]

Please mark the review comments 'resolved' once you've addressed them so the next reviewer doesn't think they're still open.

[IvanRenison]

I like the idea!
Maybe we should have StarGraph in a separate graph

[IvanRenison]

Suggested change

	lemma starGraph_adj {r x y : V} : (starGraph r).Adj x y ↔ x ≠ y ∧ (x = r ∨ y = r) := by
	unfold starGraph
	simp [SimpleGraph.fromRel]
	lemma starGraph_adj {r x y : V} : (starGraph r).Adj x y ↔ x ≠ y ∧ (x = r ∨ y = r) := by
	simp [starGraph, fromRel]

[IvanRenison]

Can you also add (starGraph r).Adj v r?

[IvanRenison]

Suggested change

	lemma starGraph_not_center_imp_degree_one [Fintype V] [DecidableEq V] {r v : V} (h : v ≠ r) :
	(starGraph r).degree v = 1 :=
	lemma degree_starGraph_of_ne_center [Fintype V] [DecidableEq V] {r v : V} (h : v ≠ r) :
	(starGraph r).degree v = 1 :=

[IvanRenison]

Suggested change

	lemma starGraph_center_degree [Fintype V] [DecidableEq V] {r : V} :
	(starGraph r).degree r = Fintype.card V - 1 := by
	lemma degree_starGraph_center [Fintype V] [DecidableEq V] {r : V} :
	(starGraph r).degree r = Fintype.card V - 1 := by

[SnirBroshi]

Suggested change

	SimpleGraph.fromRel (fun x _ => x = r)
	.fromRel fun v _ ↦ v = r

[SnirBroshi]

How about moving this to a new file that imports Acyclic.lean?

[IvanRenison]

I suggested the same. I think it is a good idea to have it in a new file

[SnirBroshi]

Hey @JovanGerb, should this be inferInstanceAs?

(I'm not sure what the rules are)

[JovanGerb]

Yes! We shouldn't use tactics like unfold or dsimp to create instances.

(with Decidable instances we typically don't run into diamonds, so it would probably not have been a problem in this case. But it's better to consistently apply these rules)