Theorems · Theorem · combinatorics
SimpleGraph.connected_iff_diam_ne_zero
∀ {α : Type u_1} {G : SimpleGraph α} [Finite α] [Nontrivial α], G.Connected ↔ G.diam ≠ 0A finite and nontrivial graph is connected if and only if its diameter is not zero.
See also connected_iff_ediam_ne_top for the extended diameter version.
- Defined in
- Mathlib.Combinatorics.SimpleGraph.Diam
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FiniteNontrivial
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Cites10
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Top.topproof · cited by 9,680
- SimpleGraphstatement and proof · cited by 3,072
- Finitestatement and proof · cited by 3,029
- Nontrivialstatement and proof · cited by 2,416
- not_iff_notproof · cited by 159
- SimpleGraph.Connectedstatement · cited by 75
- SimpleGraph.ediamproof · cited by 44
- SimpleGraph.diamstatement and proof · cited by 15
- SimpleGraph.connected_iff_ediam_ne_topproof · cited by 1
- SimpleGraph.diam_eq_zero_iff_ediam_eq_topproof · cited by 1
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