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Theorem List for Metamath Proof Explorer - 26801-26900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwwlksnextprop 26801* Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)})

Theoremdisjxwwlkn 26802* Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}

Theoremhashwwlksnext 26803* Number of walks (as words) extended by an edge as a sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       ((Vtx‘𝐺) ∈ Fin → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}) = Σ𝑦𝑌 (#‘{𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}))

Theoremwwlksnwwlksnon 26804* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))

Theoremwspthsnwspthsnon 26805* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏)))

Theoremwwlksnon0 26806 Conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)

Theoremwspthsnonn0vne 26807 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) ≠ ∅) → 𝑋𝑌)

Theoremwspthsswwlkn 26808 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)

Theoremwspthnfi 26809 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)

Theoremwwlksnonfi 26810 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.)
((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ Fin)

Theoremwspthsswwlknon 26811 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)

Theoremwspthnonfi 26812 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)

Theoremwspniunwspnon 26813* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ ∧ 𝐺𝑈) → (𝑁 WSPathsN 𝐺) = 𝑥𝑉 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))

Theoremwspn0 26814 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.)
𝑉 = (Vtx‘𝐺)       (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅)

16.3.8  Walks/paths of length 2 (as length 3 strings)

Theorem2wlkdlem1 26815 Lemma 1 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (#‘𝑃) = ((#‘𝐹) + 1)

Theorem2wlkdlem2 26816 Lemma 2 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (0..^(#‘𝐹)) = {0, 1}

Theorem2wlkdlem3 26817 Lemma 3 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))

Theorem2wlkdlem4 26818* Lemma 4 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ∀𝑘 ∈ (0...(#‘𝐹))(𝑃𝑘) ∈ 𝑉)

Theorem2wlkdlem5 26819* Lemma 5 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theorem2pthdlem1 26820* Lemma 1 for 2pthd 26830. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝑃))∀𝑗 ∈ (1..^(#‘𝐹))(𝑘𝑗 → (𝑃𝑘) ≠ (𝑃𝑗)))

Theorem2wlkdlem6 26821 Lemma 6 for 2wlkd 26826. (Contributed by AV, 23-Jan-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))

Theorem2wlkdlem7 26822 Lemma 7 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))

Theorem2wlkdlem8 26823 Lemma 8 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾))

Theorem2wlkdlem9 26824 Lemma 9 for 2wlkd 26826. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1))))

Theorem2wlkdlem10 26825* Lemma 10 for 3wlkd 27023. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))

Theorem2wlkd 26826 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Walks‘𝐺)𝑃)

Theorem2wlkond 26827 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Theorem2trld 26828 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(Trails‘𝐺)𝑃)

Theorem2trlond 26829 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃)

Theorem2pthd 26830 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(Paths‘𝐺)𝑃)

Theorem2spthd 26831 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)    &   (𝜑𝐴𝐶)       (𝜑𝐹(SPaths‘𝐺)𝑃)

Theorem2pthond 26832 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)    &   (𝜑𝐴𝐶)       (𝜑𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃)

Theorem2pthon3v 26833* 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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) ∧ (𝐴𝐵𝐴𝐶𝐵𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2))

Theoremumgr2adedgwlklem 26834 Lemma for umgr2adedgwlk 26835, umgr2adedgspth 26838, 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) → ((𝐴𝐵𝐵𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))

Theoremumgr2adedgwlk 26835 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”⟩    &   (𝜑𝐺 ∈ UMGraph )    &   (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})    &   (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})       (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))

Theoremumgr2adedgwlkon 26836 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”⟩    &   (𝜑𝐺 ∈ UMGraph )    &   (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})    &   (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Theoremumgr2adedgwlkonALT 26837 Alternate proof for umgr2adedgwlkon 26836, using umgr2adedgwlk 26835, 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”⟩    &   (𝜑𝐺 ∈ UMGraph )    &   (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})    &   (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)

Theoremumgr2adedgspth 26838 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”⟩    &   (𝜑𝐺 ∈ UMGraph )    &   (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   (𝜑 → (𝐼𝐽) = {𝐴, 𝐵})    &   (𝜑 → (𝐼𝐾) = {𝐵, 𝐶})    &   (𝜑𝐴𝐶)       (𝜑𝐹(SPaths‘𝐺)𝑃)

Theoremumgr2wlk 26839* 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) → ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))

Theoremumgr2wlkon 26840* 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) → ∃𝑓𝑝 𝑓(𝐴(WalksOn‘𝐺)𝐶)𝑝)

Theoremwwlks2onv 26841 If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
𝑉 = (Vtx‘𝐺)       ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉))

Theoremelwwlks2ons3 26842* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑈𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))

Theorems3wwlks2on 26843* 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 ∧ 𝐴𝑉𝐶𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘𝑓) = 2)))

Theoremumgrwwlks2on 26844 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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))

Theoremwwlks2onsym 26845 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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐶𝐵𝐴”⟩ ∈ (𝐶(2 WWalksNOn 𝐺)𝐴)))

Theoremelwwlks2on 26846* 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 ∧ 𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))

Theoremelwspths2on 26847* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

Theoremwpthswwlks2on 26848 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.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Theorem2wspdisj 26849* 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)

Theorem2wspiundisj 26850* 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 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)

Theoremusgr2wspthons3 26851 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))

Theoremusgr2wspthon 26852* 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 ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 ((𝑇 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝐴𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))))

Theoremelwwlks2s3 26853* A walk of length 2 between two vertices as length 3 string is a length 3 string. (Contributed by AV, 18-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Theoremmidwwlks2s3 26854* 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) → ∃𝑏𝑉 (𝑊‘1) = 𝑏)

Theoremelwwlks2 26855* 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.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Theoremelwspths2spth 26856* 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.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

16.3.9  Walks in regular graphs

Theoremrusgrnumwwlkl1 26857* 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 𝐾𝑃𝑉) → (#‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Theoremrusgrnumwwlkslem 26858* Lemma for rusgrnumwwlks 26863. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
(𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})

Theoremrusgrnumwwlklem 26859* Lemma for rusgrnumwwlk 26864 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝑃𝑉𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))

Theoremrusgrnumwwlkb0 26860* Induction base 0 for rusgrnumwwlk 26864. 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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉) → (𝑃𝐿0) = 1)

Theoremrusgrnumwwlkb1 26861* Induction base 1 for rusgrnumwwlk 26864. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (𝑃𝐿1) = 𝐾)

Theoremrusgr0edg 26862* 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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺 RegUSGraph 0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)

Theoremrusgrnumwwlks 26863* Induction step for rusgrnumwwlk 26864. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))

Theoremrusgrnumwwlk 26864* 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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))

Theoremrusgrnumwwlkg 26865* 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 26864. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁))

Theoremrusgrnumwlkg 26866* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))

Theoremclwwlknclwwlkdifs 26867 The set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between the set of walks of length n starting with this vertex and the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}       𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)

Theoremclwwlknclwwlkdifnum 26868* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}    &   𝑉 = (Vtx‘𝐺)       (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾𝑁) − (#‘𝐵)))

16.3.10  Closed walks as words

In general, a closed walk is an alternating sequence of vertices and edges, as defined in df-clwlks 26661: 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 cyles (which are special closed walks) in [Diestel] p. 7. As for "walks as words" in general, the concept of a Word, see df-word 13294, is also used in definitions df-clwwlks 26871 and df-clwwlksn 26872, 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-clwwlks 26871 and df-clwwlksn 26872, 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.

Notice that by this definition, a single vertex cannot be represented as closed walk, since the word ⟨“𝑣”⟩ with vertex v represents the walk "𝑣 𝑣", which is a (closed) walk of length 1 (if there is an edge/loop from 𝑣 to 𝑣). Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 26898. This is also the reason for defining the set ClWWalksN of all closed walks of a fixed length as words over the set of vertices as function over and not 0, because (0 ClWWalksN 𝐺) = ∅ (see clwwlksn0 26900) would hold anyway. In all other cases, however, a closed walk (of length at least 1) corresponds to a closed walk as word, at least in a simple pseudograph, see clwlkclwwlk2 26898.

Syntaxcclwwlks 26869 Extend class notation with closed walks (in an undirected graph) as word over the set of vertices.
class ClWWalks

Syntaxcclwwlksn 26870 Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices.
class ClWWalksN

Definitiondf-clwwlks 26871* 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 26661. 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)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})

Definitiondf-clwwlksn 26872* Define the set of all closed walks (in an undirected graph) of a fixed length n 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 26661. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
ClWWalksN = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛})

Theoremclwwlks 26873* 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)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ 𝐸)}

Theoremisclwwlks 26874* 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 𝑉𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸))

Theoremclwwlksn 26875* The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
(𝑁 ∈ ℕ → (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})

Theoremisclwwlksn 26876 A word over the set of vertices representing a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)
(𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁)))

Theoremclwwlkbp 26877 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 𝑉𝑊 ≠ ∅))

Theoremclwwlknbp0 26878 Basic properties of a closed walk of a fixed length as word. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 24-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝐺 ∈ V ∧ 𝑁 ∈ ℕ) ∧ (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁)))

Theoremclwwlknbp 26879 Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁))

Theoremclwwlksnwrd 26880 A closed walk of a fixed length as word is a word over the vertices. (Contributed by AV, 30-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word 𝑉)

Theoremclwwlknp 26881* Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 24-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸))

Theoremisclwwlksng 26882 Properties of a word to represent a closed walk of a fixed length. Generalization of isclwwlksn 26876. (Contributed by AV, 25-Apr-2021.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁))

Theoremisclwwlksnx 26883* Properties of a word to represent a closed walk of a fixed length , definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁)))

Theoremclwwlksnndef 26884 Conditions for ClWWalksN not being defined. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)
((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅)

Theoremclwwlkclwwlkn 26885 A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)
(𝑃 ∈ (𝑁 ClWWalksN 𝐺) → 𝑃 ∈ (ClWWalks‘𝐺))

Theoremclwwlkssclwwlksn 26886 The closed walks of a fixed length as words are closed walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 12-Apr-2021.)
(𝑁 ClWWalksN 𝐺) ⊆ (ClWWalks‘𝐺)

Theoremclwlkclwwlklem2a1 26887* Lemma 1 for clwlkclwwlklem2a 26893. (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)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))

Theoremclwlkclwwlklem2a2 26888* Lemma 2 for clwlkclwwlklem2a 26893. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))

Theoremclwlkclwwlklem2a3 26889* Lemma 3 for clwlkclwwlklem2a 26893. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (𝑃‘(#‘𝐹)) = ( lastS ‘𝑃))

Theoremclwlkclwwlklem2fv1 26890* Lemma 4a for clwlkclwwlklem2a 26893. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       (((#‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((#‘𝑃) − 2))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))

Theoremclwlkclwwlklem2fv2 26891* Lemma 4b for clwlkclwwlklem2a 26893. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       (((#‘𝑃) ∈ ℕ0 ∧ 2 ≤ (#‘𝑃)) → (𝐹‘((#‘𝑃) − 2)) = (𝐸‘{(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)}))

Theoremclwlkclwwlklem2a4 26892* Lemma 4 for clwlkclwwlklem2a 26893. (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‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝐸:dom 𝐸1-1𝑅𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((#‘𝑃) − 1))) → ({(𝑃𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹𝐼)) = {(𝑃𝐼), (𝑃‘(𝐼 + 1))})))

Theoremclwlkclwwlklem2a 26893* Lemma for clwlkclwwlklem2 26895. (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‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝐸:dom 𝐸1-1𝑅𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremclwlkclwwlklem1 26894* Lemma 1 for clwlkclwwlk 26897. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
((𝐸:dom 𝐸1-1𝑅𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))))

Theoremclwlkclwwlklem2 26895* Lemma 2 for clwlkclwwlk 26897. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
(((𝐸:dom 𝐸1-1𝑅𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ (∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐹) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸))

Theoremclwlkclwwlklem3 26896* Lemma 3 for clwlkclwwlk 26897. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
((𝐸:dom 𝐸1-1𝑅𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))

Theoremclwlkclwwlk 26897* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)𝑃 ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (𝑃 substr ⟨0, ((#‘𝑃) − 1)⟩) ∈ (ClWWalks‘𝐺))))

Theoremclwlkclwwlk2 26898* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))

Theoremclwwlksgt0 26899 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))

Theoremclwwlksn0 26900 There is no closed walk of length 0 (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.)
(0 ClWWalksN 𝐺) = ∅

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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