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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwspthsswwlknon 27301 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 27302 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 27303* 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 ∧ 𝐺𝑈) → (𝑁 WSPathsN 𝐺) = 𝑥𝑉 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))

Theoremwspn0 27304 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) = ∅)

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

Theorem2wlkdlem1 27305 Lemma 1 for 2wlkd 27316. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (♯‘𝑃) = ((♯‘𝐹) + 1)

Theorem2wlkdlem2 27306 Lemma 2 for 2wlkd 27316. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (0..^(♯‘𝐹)) = {0, 1}

Theorem2wlkdlem3 27307 Lemma 3 for 2wlkd 27316. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))

Theorem2wlkdlem4 27308* Lemma 4 for 2wlkd 27316. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃𝑘) ∈ 𝑉)

Theorem2wlkdlem5 27309* Lemma 5 for 2wlkd 27316. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theorem2pthdlem1 27310* Lemma 1 for 2pthd 27320. (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 27311 Lemma 6 for 2wlkd 27316. (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 27312 Lemma 7 for 2wlkd 27316. (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 27313 Lemma 8 for 2wlkd 27316. (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 27314 Lemma 9 for 2wlkd 27316. (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 27315* Lemma 10 for 3wlkd 27573. (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 27316 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 27317 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 27318 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 27319 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 27320 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 27321 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 27322 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 27323* 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 27324 Lemma for umgr2adedgwlk 27325, umgr2adedgspth 27328, 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 27325 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 27326 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 27327 Alternate proof for umgr2adedgwlkon 27326, using umgr2adedgwlk 27325, 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 27328 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 27329* 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 27330* 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)

Theoremelwwlks2s3 27331* 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) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)

Theoremmidwwlks2s3 27332* 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)

Theoremwwlks2onv 27333 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)       ((𝐵𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴𝑉𝐵𝑉𝐶𝑉))

Theoremelwwlks2ons3im 27334 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))

Theoremelwwlks2ons3 27335* 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) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))

Theorems3wwlks2on 27336* 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 27337 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 27338 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 27339* 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 27340* 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 ∧ 𝐴𝑉𝐶𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))

Theoremwpthswwlks2on 27341 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 ∧ 𝐴𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))

Theorem2wspdisj 27342* 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 27343* 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 27344 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 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))

Theoremusgr2wspthon 27345* 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))))

Theoremelwwlks2 27346* 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) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (♯‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Theoremelwspths2spth 27347* 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) ↔ ∃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 27348* 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)       ((𝐺RegUSGraph𝐾𝑃𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)

Theoremrusgrnumwwlkslem 27349* Lemma for rusgrnumwwlks 27354. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
(𝑌 ∈ {𝑤𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤𝑋 ∣ (𝜑𝜓)} = {𝑤𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃𝜓)})

Theoremrusgrnumwwlklem 27350* Lemma for rusgrnumwwlk 27356 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 27351* Induction base 0 for rusgrnumwwlk 27356. 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 27352* Induction base 1 for rusgrnumwwlk 27356. (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}))       ((𝐺RegUSGraph𝐾𝑃𝑉) → (𝑃𝐿1) = 𝐾)

Theoremrusgr0edg 27353* 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}))       ((𝐺RegUSGraph0 ∧ 𝑃𝑉𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)

Theoremrusgrnumwwlks 27354* Induction step for rusgrnumwwlk 27356. (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 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))

TheoremrusgrnumwwlksOLD 27355* Obsolete proof of rusgrnumwwlks 27354 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐿 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))       ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))

Theoremrusgrnumwwlk 27356* 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}))       ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾𝑁))

Theoremrusgrnumwwlkg 27357* 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 27356. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾𝑁))

Theoremrusgrnumwlkg 27358* 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)       ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃𝑉𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑃)}) = (𝐾𝑁))

Theoremclwwlknclwwlkdif 27359* 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 = (𝐶𝐵)

Theoremclwwlknclwwlkdifnum 27360* 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)       (((𝐺RegUSGraph𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (♯‘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 27123: 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 13600, is also used in definitions df-clwwlk 27362 and df-clwwlkn 27414, 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 27362, df-clwwlkn 27414 and df-clwwlknon 27490, 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 27534, 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 27432.

Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 27384 or clwlkclwwlken 27400. Although the set ClWWalksN of all closed walks of a fixed length as words over the set of vertices is defined as function over 0, the fixed length is usually not 0, because (0 ClWWalksN 𝐺) = ∅ (see clwwlkn0 27417).

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 27490. This definition is also based on 0 instead of , with (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 27494). clwwlknon1le1 27503 states that there is at most one (closed) walk of length 1 on a vertex, which would consist of a loop (see clwwlknon1loop 27500). And in a 𝐾-regular graph, there are 𝐾 closed walks of length 2 on each vertex, see clwwlknon2num 27507.

16.3.10.1  Closed walks as words

Syntaxcclwwlk 27361 Extend class notation with closed walks (in an undirected graph) as word over the set of vertices.
class ClWWalks

Definitiondf-clwwlk 27362* 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 27123. 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))})

Theoremclwwlk 27363* 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)}

Theoremisclwwlk 27364* 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))

Theoremclwwlkbp 27365 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 ≠ ∅))

Theoremclwwlkgt0 27366 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))

Theoremclwwlksswrd 27367 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)

Theoremclwwlk1loop 27368 A closed walk of length 1 is a loop. See also clwlkl1loop 27135. (Contributed by AV, 24-Apr-2021.)
((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))

Theoremclwwlkccatlem 27369* Lemma for clwwlkccat 27370: 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)){(𝐴𝑖), (𝐴‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ≠ ∅) ∧ ∀𝑗 ∈ (0..^((♯‘𝐵) − 1)){(𝐵𝑗), (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))

Theoremclwwlkccat 27370 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))

Theoremumgrclwwlkge2 27371 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)))

Theoremclwlkclwwlklem2a1 27372* Lemma 1 for clwlkclwwlklem2a 27378. (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 27373* Lemma 2 for clwlkclwwlklem2a 27378. (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 27374* Lemma 3 for clwlkclwwlklem2a 27378. (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 27375* Lemma 4a for clwlkclwwlklem2a 27378. (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 27376* Lemma 4b for clwlkclwwlklem2a 27378. (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 27377* Lemma 4 for clwlkclwwlklem2a 27378. (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 27378* Lemma for clwlkclwwlklem2 27380. (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 27379* Lemma 1 for clwlkclwwlk 27382. (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 27380* Lemma 2 for clwlkclwwlk 27382. (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 27381* Lemma 3 for clwlkclwwlk 27382. (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 27382* 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))))

TheoremclwlkclwwlkOLD 27383* Obsolete version of clwlkclwwlk 27382 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
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 27384* 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)))

Theoremclwlkclwwlk2OLD 27385* Obsolete proof of clwlkclwwlk2 27384 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 7-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))

Theoremclwlkclwwlkflem 27386* Lemma for clwlkclwwlkf 27396. (Contributed by AV, 24-May-2022.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐴 = (1st𝑈)    &   𝐵 = (2nd𝑈)       (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Theoremclwlkclwwlkf1lem2 27387* Lemma 2 for clwlkclwwlkf1 27398. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐴 = (1st𝑈)    &   𝐵 = (2nd𝑈)    &   𝐷 = (1st𝑊)    &   𝐸 = (2nd𝑊)       ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)))

Theoremclwlkclwwlkf1lem2OLD 27388* Obsolete version of clwlkclwwlkf1lem2 27387 as of 12-Oct-2022. (Contributed by AV, 24-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐴 = (1st𝑈)    &   𝐵 = (2nd𝑈)    &   𝐷 = (1st𝑊)    &   𝐸 = (2nd𝑊)       ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖)))

Theoremclwlkclwwlkf1lem3 27389* Lemma 3 for clwlkclwwlkf1 27398. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐴 = (1st𝑈)    &   𝐵 = (2nd𝑈)    &   𝐷 = (1st𝑊)    &   𝐸 = (2nd𝑊)       ((𝑈𝐶𝑊𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))

Theoremclwlkclwwlkf1lem3OLD 27390* Obsolete version of clwlkclwwlkf1lem3 27389 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐴 = (1st𝑈)    &   𝐵 = (2nd𝑈)    &   𝐷 = (1st𝑊)    &   𝐸 = (2nd𝑊)       ((𝑈𝐶𝑊𝐶 ∧ (𝐵 substr ⟨0, (♯‘𝐴)⟩) = (𝐸 substr ⟨0, (♯‘𝐷)⟩)) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵𝑖) = (𝐸𝑖))

Theoremclwlkclwwlkfolem 27391* Lemma for clwlkclwwlkfo 27397. (Contributed by AV, 25-May-2022.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}       ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)

TheoremclwlkclwwlkfOLD 27392* Obsolete version of clwlkclwwlkf 27396 as of 12-Oct-2022. (Contributed by AV, 23-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))       (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

TheoremclwlkclwwlkfoOLD 27393* Obsolete version of clwlkclwwlkfo 27397 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Revised by AV, 25-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))       (𝐺 ∈ USPGraph → 𝐹:𝐶onto→(ClWWalks‘𝐺))

Theoremclwlkclwwlkf1OLD 27394* Obsolete version of clwlkclwwlkf1 27398 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))       (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))

Theoremclwlkclwwlkf1oOLD 27395* Obsolete version of clwlkclwwlkf1o 27399 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 24-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) substr ⟨0, ((♯‘(2nd𝑐)) − 1)⟩))       (𝐺 ∈ USPGraph → 𝐹:𝐶1-1-onto→(ClWWalks‘𝐺))

Theoremclwlkclwwlkf 27396* 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 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))       (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Theoremclwlkclwwlkfo 27397* 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 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))       (𝐺 ∈ USPGraph → 𝐹:𝐶onto→(ClWWalks‘𝐺))

Theoremclwlkclwwlkf1 27398* 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 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))       (𝐺 ∈ USPGraph → 𝐹:𝐶1-1→(ClWWalks‘𝐺))

Theoremclwlkclwwlkf1o 27399* 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 ≤ (♯‘(1st𝑤))}    &   𝐹 = (𝑐𝐶 ↦ ((2nd𝑐) prefix ((♯‘(2nd𝑐)) − 1)))       (𝐺 ∈ USPGraph → 𝐹:𝐶1-1-onto→(ClWWalks‘𝐺))

Theoremclwlkclwwlken 27400* 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 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))

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