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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | wwlksnwwlksnon 27701* | A walk of fixed length is a walk of fixed length between two vertices. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 13-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)) | ||
Theorem | wspthsnwspthsnon 27702* | A simple path of fixed length is a simple path of fixed length between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-May-2021.) (Revised by AV, 13-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏)) | ||
Theorem | wspthsnonn0vne 27703 | If the set of simple paths of length at least 1 between two vertices is not empty, the two vertices must be different. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ (𝑋(𝑁 WSPathsNOn 𝐺)𝑌) ≠ ∅) → 𝑋 ≠ 𝑌) | ||
Theorem | wspthsswwlkn 27704 | The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.) |
⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) | ||
Theorem | wspthnfi 27705 | In a finite graph, the set of simple paths of a fixed length is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 18-May-2021.) |
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WSPathsN 𝐺) ∈ Fin) | ||
Theorem | wwlksnonfi 27706 | In a finite graph, the set of walks of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.) (Proof shortened by AV, 15-Mar-2022.) |
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ Fin) | ||
Theorem | wspthsswwlknon 27707 | The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length between the two vertices. (Contributed by AV, 15-May-2021.) |
⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ⊆ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) | ||
Theorem | wspthnonfi 27708 | In a finite graph, the set of simple paths of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.) |
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∈ Fin) | ||
Theorem | wspniunwspnon 27709* | The set of nonempty simple paths of fixed length is the double union of the simple paths of the fixed length between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 15-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦)) | ||
Theorem | wspn0 27710 | If there are no vertices, then there are no simple paths (of any length), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 13-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅) | ||
Theorem | 2wlkdlem1 27711 | Lemma 1 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 ⇒ ⊢ (♯‘𝑃) = ((♯‘𝐹) + 1) | ||
Theorem | 2wlkdlem2 27712 | Lemma 2 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 ⇒ ⊢ (0..^(♯‘𝐹)) = {0, 1} | ||
Theorem | 2wlkdlem3 27713 | Lemma 3 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ⇒ ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶)) | ||
Theorem | 2wlkdlem4 27714* | Lemma 4 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉) | ||
Theorem | 2wlkdlem5 27715* | Lemma 5 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) | ||
Theorem | 2pthdlem1 27716* | Lemma 1 for 2pthd 27726. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗))) | ||
Theorem | 2wlkdlem6 27717 | Lemma 6 for 2wlkd 27722. (Contributed by AV, 23-Jan-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) ⇒ ⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾))) | ||
Theorem | 2wlkdlem7 27718 | Lemma 7 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) ⇒ ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V)) | ||
Theorem | 2wlkdlem8 27719 | Lemma 8 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) ⇒ ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾)) | ||
Theorem | 2wlkdlem9 27720 | Lemma 9 for 2wlkd 27722. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) ⇒ ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1)))) | ||
Theorem | 2wlkdlem10 27721* | Lemma 10 for 3wlkd 27955. (Contributed by AV, 14-Feb-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) | ||
Theorem | 2wlkd 27722 | Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.) (Revised by AV, 23-Jan-2021.) (Proof shortened by AV, 14-Feb-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃) | ||
Theorem | 2wlkond 27723 | A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) ⇒ ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) | ||
Theorem | 2trld 27724 | Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) ⇒ ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | ||
Theorem | 2trlond 27725 | A trail of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) ⇒ ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃) | ||
Theorem | 2pthd 27726 | A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) ⇒ ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃) | ||
Theorem | 2spthd 27727 | A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) | ||
Theorem | 2pthond 27728 | A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Proof shortened by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.) |
⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾))) & ⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃) | ||
Theorem | 2pthon3v 27729* | For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (♯‘𝑓) = 2)) | ||
Theorem | umgr2adedgwlklem 27730 | Lemma for umgr2adedgwlk 27731, umgr2adedgspth 27734, etc. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))) | ||
Theorem | umgr2adedgwlk 27731 | In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) & ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) & ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2)))) | ||
Theorem | umgr2adedgwlkon 27732 | In a multigraph, two adjacent edges form a walk between two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) & ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) & ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) | ||
Theorem | umgr2adedgwlkonALT 27733 | Alternate proof for umgr2adedgwlkon 27732, using umgr2adedgwlk 27731, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) & ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) & ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) ⇒ ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃) | ||
Theorem | umgr2adedgspth 27734 | In a multigraph, two adjacent edges with different endvertices form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) & ⊢ 𝐼 = (iEdg‘𝐺) & ⊢ 𝐹 = 〈“𝐽𝐾”〉 & ⊢ 𝑃 = 〈“𝐴𝐵𝐶”〉 & ⊢ (𝜑 → 𝐺 ∈ UMGraph) & ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) & ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵}) & ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶}) & ⊢ (𝜑 → 𝐴 ≠ 𝐶) ⇒ ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃) | ||
Theorem | umgr2wlk 27735* | In a multigraph, there is a walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))) | ||
Theorem | umgr2wlkon 27736* | For each pair of adjacent edges in a multigraph, there is a walk of length 2 between the not common vertices of the edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) |
⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝 𝑓(𝐴(WalksOn‘𝐺)𝐶)𝑝) | ||
Theorem | elwwlks2s3 27737* | A walk of length 2 as word is a length 3 string. (Contributed by AV, 18-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = 〈“𝑎𝑏𝑐”〉) | ||
Theorem | midwwlks2s3 27738* | There is a vertex between the endpoints of a walk of length 2 between two vertices as length 3 string. (Contributed by AV, 10-Jan-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑏 ∈ 𝑉 (𝑊‘1) = 𝑏) | ||
Theorem | wwlks2onv 27739 | If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Proof shortened by AV, 14-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐵 ∈ 𝑈 ∧ 〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) | ||
Theorem | elwwlks2ons3im 27740 | A walk as word of length 2 between two vertices is a length 3 string and its second symbol is a vertex. (Contributed by AV, 14-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = 〈“𝐴(𝑊‘1)𝐶”〉 ∧ (𝑊‘1) ∈ 𝑉)) | ||
Theorem | elwwlks2ons3 27741* | For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))) | ||
Theorem | s3wwlks2on 27742* | A length 3 string which represents a walk of length 2 between two vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)〈“𝐴𝐵𝐶”〉 ∧ (♯‘𝑓) = 2))) | ||
Theorem | umgrwwlks2on 27743 | A walk of length 2 between two vertices as word in a multigraph. This theorem would also hold for pseudographs, but to prove this the cases 𝐴 = 𝐵 and/or 𝐵 = 𝐶 must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | ||
Theorem | wwlks2onsym 27744 | There is a walk of length 2 from one vertex to another vertex iff there is a walk of length 2 from the other vertex to the first vertex. (Contributed by AV, 7-Jan-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ 〈“𝐶𝐵𝐴”〉 ∈ (𝐶(2 WWalksNOn 𝐺)𝐴))) | ||
Theorem | elwwlks2on 27745* | A walk of length 2 between two vertices as length 3 string. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (♯‘𝑓) = 2)))) | ||
Theorem | elwspths2on 27746* | A simple path of length 2 between two vertices (in a graph) as length 3 string. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 16-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = 〈“𝐴𝑏𝐶”〉 ∧ 〈“𝐴𝑏𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶)))) | ||
Theorem | wpthswwlks2on 27747 | For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.) |
⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵)) | ||
Theorem | 2wspdisj 27748* | All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 9-Jan-2022.) |
⊢ Disj 𝑏 ∈ (𝑉 ∖ {𝐴})(𝐴(2 WSPathsNOn 𝐺)𝑏) | ||
Theorem | 2wspiundisj 27749* | All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.) (Revised by AV, 14-May-2021.) (Proof shortened by AV, 9-Jan-2022.) |
⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏) | ||
Theorem | usgr2wspthons3 27750 | A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (〈“𝐴𝐵𝐶”〉 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))) | ||
Theorem | usgr2wspthon 27751* | A simple path of length 2 between two vertices corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = 〈“𝐴𝑏𝐶”〉 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸)))) | ||
Theorem | elwwlks2 27752* | A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 14-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = 〈“𝑎𝑏𝑐”〉 ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))) | ||
Theorem | elwspths2spth 27753* | A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 16-Mar-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = 〈“𝑎𝑏𝑐”〉 ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))) | ||
Theorem | rusgrnumwwlkl1 27754* | In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾) | ||
Theorem | rusgrnumwwlkslem 27755* | Lemma for rusgrnumwwlks 27760. (Contributed by Alexander van der Vekens, 23-Aug-2018.) |
⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)}) | ||
Theorem | rusgrnumwwlklem 27756* | Lemma for rusgrnumwwlk 27761 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})) | ||
Theorem | rusgrnumwwlkb0 27757* | Induction base 0 for rusgrnumwwlk 27761. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1) | ||
Theorem | rusgrnumwwlkb1 27758* | Induction base 1 for rusgrnumwwlk 27761. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿1) = 𝐾) | ||
Theorem | rusgr0edg 27759* | Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0) | ||
Theorem | rusgrnumwwlks 27760* | Induction step for rusgrnumwwlk 27761. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 27-May-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1)))) | ||
Theorem | rusgrnumwwlk 27761* |
In a 𝐾-regular graph, the number of walks
of a fixed length 𝑁
from a fixed vertex is 𝐾 to the power of 𝑁. By
definition,
(𝑁
WWalksN 𝐺) is the
set of walks (as words) with length 𝑁,
and (𝑃𝐿𝑁) is the number of walks with length
𝑁
starting at
the vertex 𝑃. Because of the 𝐾-regularity, a walk can be
continued in 𝐾 different ways at the end vertex of
the walk, and
this repeated 𝑁 times.
This theorem even holds for 𝑁 = 0: in this case, the walk consists of only one vertex 𝑃, so the number of walks of length 𝑁 = 0 starting with 𝑃 is (𝐾↑0) = 1. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁)) | ||
Theorem | rusgrnumwwlkg 27762* | In a 𝐾-regular graph, the number of walks (as words) of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. Closed form of rusgrnumwwlk 27761. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁)) | ||
Theorem | rusgrnumwlkg 27763* | In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular." This theorem even holds for n=0: then the walk consists of only one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 5-Aug-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁)) | ||
Theorem | clwwlknclwwlkdif 27764* | The set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑉 and ending not at this vertex is the difference between the set 𝐶 of walks of length 𝑁 starting with this vertex 𝑋 and the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.) |
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} & ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋) & ⊢ 𝐶 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ⇒ ⊢ 𝐴 = (𝐶 ∖ 𝐵) | ||
Theorem | clwwlknclwwlkdifnum 27765* | In a 𝐾-regular graph, the size of the set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 8-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.) |
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)} & ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋) & ⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘𝐴) = ((𝐾↑𝑁) − (♯‘𝐵))) | ||
In general, a closed walk is an alternating sequence of vertices and edges, as defined in df-clwlks 27560: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n), with p(n) = p(0). Often, it is sufficient to refer to a walk by the (cyclic) sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(0), see the corresponding remark on cycles (which are special closed walks) in [Diestel] p. 7. As for "walks as words" in general, the concept of a Word, see df-word 13858, is also used in definitions df-clwwlk 27767 and df-clwwlkn 27810, and the representation of a closed walk as the sequence of its vertices is called "closed walk as word". In contrast to "walks as words", the terminating vertex p(n) of a closed walk is omitted in the representation of a closed walk as word, see definitions df-clwwlk 27767, df-clwwlkn 27810 and df-clwwlknon 27873, because it is always equal to the first vertex of the closed walk. This represenation has the advantage that the vertices can be cyclically shifted without changing the represented closed walk. Furthermore, the length of a closed walk (i.e. the number of its edges) equals the number of symbols/vertices of the word representing the closed walk. To avoid to handle the degenerate case of representing a (closed) walk of length 0 by the empty word, this case is excluded within the definition (𝑤 ≠ ∅). This is because a walk of length 0 is anchored at an arbitrary vertex by the general definition for closed walks, see 0clwlkv 27916, which neither can be reflected by the empty word nor by a singleton word 〈“𝑣”〉 with vertex v : 〈“𝑣”〉 represents the walk "𝑣 𝑣", which is a (closed) walk of length 1 (if there is an edge/loop from 𝑣 to 𝑣), see loopclwwlkn1b 27827. Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 27788 or clwlkclwwlken 27797. Although the set ClWWalksN of all closed walks of a fixed length as words over the set of vertices is defined as function over ℕ0, the fixed length is usually not 0, because (0 ClWWalksN 𝐺) = ∅ (see clwwlkn0 27813). Analogous to (𝐴(𝑁 WWalksNOn 𝐺)𝐵), the set of walks of a fixed length 𝑁 between two vertices 𝐴 and 𝐵, the set (𝑋(ClWWalksNOn‘𝐺)𝑁) of closed walks of a fixed length 𝑁 anchored at a fixed vertex 𝑋 is defined by df-clwwlknon 27873. This definition is also based on ℕ0 instead of ℕ, with (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 27877). clwwlknon1le1 27886 states that there is at most one (closed) walk of length 1 on a vertex, which would consist of a loop (see clwwlknon1loop 27883). And in a 𝐾-regular graph, there are 𝐾 closed walks of length 2 on each vertex, see clwwlknon2num 27890. | ||
Syntax | cclwwlk 27766 | Extend class notation with closed walks (in an undirected graph) as word over the set of vertices. |
class ClWWalks | ||
Definition | df-clwwlk 27767* | Define the set of all closed walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlks 27560. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))}) | ||
Theorem | clwwlk 27768* | The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (ClWWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ 𝐸)} | ||
Theorem | isclwwlk 27769* | Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (Edg‘𝐺) ⇒ ⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸)) | ||
Theorem | clwwlkbp 27770 | Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ 𝑉 = (Vtx‘𝐺) ⇒ ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅)) | ||
Theorem | clwwlkgt0 27771 | There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑊)) | ||
Theorem | clwwlksswrd 27772 | Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) |
⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺) | ||
Theorem | clwwlk1loop 27773 | A closed walk of length 1 is a loop. See also clwlkl1loop 27572. (Contributed by AV, 24-Apr-2021.) |
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)) | ||
Theorem | clwwlkccatlem 27774* | Lemma for clwwlkccat 27775: index 𝑗 is shifted up by (♯‘𝐴), and the case 𝑖 = ((♯‘𝐴) − 1) is covered by the "bridge" {(lastS‘𝐴), (𝐵‘0)} = {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺). (Contributed by AV, 23-Apr-2022.) |
⊢ ((((𝐴 ∈ Word (Vtx‘𝐺) ∧ 𝐴 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ≠ ∅) ∧ ∀𝑗 ∈ (0..^((♯‘𝐵) − 1)){(𝐵‘𝑗), (𝐵‘(𝑗 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐵), (𝐵‘0)} ∈ (Edg‘𝐺)) ∧ (𝐴‘0) = (𝐵‘0)) → ∀𝑖 ∈ (0..^((♯‘(𝐴 ++ 𝐵)) − 1)){((𝐴 ++ 𝐵)‘𝑖), ((𝐴 ++ 𝐵)‘(𝑖 + 1))} ∈ (Edg‘𝐺)) | ||
Theorem | clwwlkccat 27775 | The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 23-Apr-2022.) |
⊢ ((𝐴 ∈ (ClWWalks‘𝐺) ∧ 𝐵 ∈ (ClWWalks‘𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ (ClWWalks‘𝐺)) | ||
Theorem | umgrclwwlkge2 27776 | A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.) |
⊢ (𝐺 ∈ UMGraph → (𝑃 ∈ (ClWWalks‘𝐺) → 2 ≤ (♯‘𝑃))) | ||
Theorem | clwlkclwwlklem2a1 27777* | Lemma 1 for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸)) | ||
Theorem | clwlkclwwlklem2a2 27778* | Lemma 2 for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 21-Jun-2018.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1)) | ||
Theorem | clwlkclwwlklem2a3 27779* | Lemma 3 for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 21-Jun-2018.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃)) | ||
Theorem | clwlkclwwlklem2fv1 27780* | Lemma 4a for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 22-Jun-2018.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 2))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})) | ||
Theorem | clwlkclwwlklem2fv2 27781* | Lemma 4b for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 22-Jun-2018.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (𝐹‘((♯‘𝑃) − 2)) = (◡𝐸‘{(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)})) | ||
Theorem | clwlkclwwlklem2a4 27782* | Lemma 4 for clwlkclwwlklem2a 27783. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → ({(𝑃‘𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹‘𝐼)) = {(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))) | ||
Theorem | clwlkclwwlklem2a 27783* | Lemma for clwlkclwwlklem2 27785. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)}))) ⇒ ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))) | ||
Theorem | clwlkclwwlklem1 27784* | Lemma 1 for clwlkclwwlk 27787. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓))))) | ||
Theorem | clwlkclwwlklem2 27785* | Lemma 2 for clwlkclwwlk 27787. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ 2 ≤ (♯‘𝑃)) ∧ (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) → ((lastS‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((♯‘𝐹) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸)) | ||
Theorem | clwlkclwwlklem3 27786* | Lemma 3 for clwlkclwwlk 27787. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.) |
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓))) ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)))) | ||
Theorem | clwlkclwwlk 27787* | A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 30-Oct-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)𝑃 ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (𝑃 prefix ((♯‘𝑃) − 1)) ∈ (ClWWalks‘𝐺)))) | ||
Theorem | clwlkclwwlk2 27788* | A closed walk corresponds to a closed walk as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 2-Nov-2022.) |
⊢ 𝑉 = (Vtx‘𝐺) & ⊢ 𝐸 = (iEdg‘𝐺) ⇒ ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ 〈“(𝑃‘0)”〉) ↔ 𝑃 ∈ (ClWWalks‘𝐺))) | ||
Theorem | clwlkclwwlkflem 27789* | Lemma for clwlkclwwlkf 27793. (Contributed by AV, 24-May-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)) | ||
Theorem | clwlkclwwlkf1lem2 27790* | Lemma 2 for clwlkclwwlkf1 27795. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) & ⊢ 𝐷 = (1st ‘𝑊) & ⊢ 𝐸 = (2nd ‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))) | ||
Theorem | clwlkclwwlkf1lem3 27791* | Lemma 3 for clwlkclwwlkf1 27795. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐴 = (1st ‘𝑈) & ⊢ 𝐵 = (2nd ‘𝑈) & ⊢ 𝐷 = (1st ‘𝑊) & ⊢ 𝐸 = (2nd ‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)) | ||
Theorem | clwlkclwwlkfolem 27792* | Lemma for clwlkclwwlkfo 27794. (Contributed by AV, 25-May-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ⇒ ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ (ClWalks‘𝐺)) → 〈𝑓, (𝑊 ++ 〈“(𝑊‘0)”〉)〉 ∈ 𝐶) | ||
Theorem | clwlkclwwlkf 27793* | 𝐹 is a function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by AV, 23-May-2022.) (Revised by AV, 29-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺)) | ||
Theorem | clwlkclwwlkfo 27794* | 𝐹 is a function from the nonempty closed walks onto the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by AV, 29-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺)) | ||
Theorem | clwlkclwwlkf1 27795* | 𝐹 is a one-to-one function from the nonempty closed walks into the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺)) | ||
Theorem | clwlkclwwlkf1o 27796* | 𝐹 is a bijection between the nonempty closed walks and the closed walks as words in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 29-Oct-2022.) |
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} & ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1))) ⇒ ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺)) | ||
Theorem | clwlkclwwlken 27797* | The set of the nonempty closed walks and the set of closed walks as word are equinumerous in a simple pseudograph. (Contributed by AV, 25-May-2022.) (Proof shortened by AV, 4-Nov-2022.) |
⊢ (𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺)) | ||
Theorem | clwwisshclwwslemlem 27798* | Lemma for clwwisshclwwslem 27799. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
⊢ (((𝐿 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅) | ||
Theorem | clwwisshclwwslem 27799* | Lemma for clwwisshclwws 27800. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.) |
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(♯‘𝑊))) → ((∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) → ∀𝑗 ∈ (0..^((♯‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ 𝐸)) | ||
Theorem | clwwisshclwws 27800 | Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.) |
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺)) |
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