Theorems · Theorem · combinatorics
SimpleGraph.connected_iff_ediam_ne_top
∀ {α : Type u_1} {G : SimpleGraph α} [Nonempty α] [Finite α], G.Connected ↔ G.ediam ≠ ⊤In a finite graph with nontrivial vertex set, the graph is connected
if and only if the extended diameter is not ⊤.
See connected_of_ediam_ne_top for one of the implications without
the finiteness assumptions
- Defined in
- Mathlib.Combinatorics.SimpleGraph.Diam
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topstatement and proof · cited by 9,680
- ENatstatement · cited by 4,985
- SimpleGraphstatement and proof · cited by 3,072
- Finitestatement and proof · cited by 3,029
- SimpleGraph.Connectedstatement and proof · cited by 75
- SimpleGraph.edistproof · cited by 55
- SimpleGraph.ediamstatement and proof · cited by 44
- SimpleGraph.Connected.preconnectedproof · cited by 28
- SimpleGraph.edist_ne_top_iff_reachableproof · cited by 4
- SimpleGraph.connected_of_ediam_ne_topproof · cited by 2
- SimpleGraph.exists_edist_eq_ediam_of_finiteproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- SimpleGraph.connected_iff_diam_ne_zeroproof · cited by 0