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Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremclwwlknwwlksnOLD 27001 Obsolete version of clwwlknwwlksn 27000 as of 22-Mar-2022. (Contributed by AV, 24-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ ℕ ∧ 𝑊 ∈ (𝑁ClWWalksNOLD𝐺)) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺))

Theoremclwwlknlbonbgr1 27002 The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022.)
((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0)))

Theoremclwwlkinwwlk 27003 If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Proof shortened by AV, 23-Mar-2022.)
(((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊𝑁) = (𝑊‘0)) → (𝑊 substr ⟨0, 𝑁⟩) ∈ (𝑁 ClWWalksN 𝐺))

Theoremclwwlkn1 27004 A closed walk of length 1 represented as word is a word consisting of 1 symbol representing a vertex connected to itself by (at least) one edge, that is, a loop. (Contributed by AV, 24-Apr-2021.) (Revised by AV, 11-Feb-2022.)
(𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((#‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))

Theoremloopclwwlkn1b 27005 The singleton word consisting of a vertex 𝑉 represents a closed walk of length 1 iff there is a loop at vertex 𝑉. (Contributed by AV, 11-Feb-2022.)
(𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ ⟨“𝑉”⟩ ∈ (1 ClWWalksN 𝐺)))

Theoremclwwlkn1loopb 27006* A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.)
(𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))

Theoremclwwlkn2 27007 A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)
(𝑊 ∈ (2 ClWWalksN 𝐺) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0), (𝑊‘1)} ∈ (Edg‘𝐺)))

Theoremclwwlknfi 27008 If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)

Theoremclwwlkel 27009* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 25-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}       ((𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ 𝐷)

Theoremclwwlkf 27010* Lemma 1 for clwwlkbij 27015: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))

Theoremclwwlkfv 27011* Lemma 2 for clwwlkbij 27015: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑊𝐷 → (𝐹𝑊) = (𝑊 substr ⟨0, 𝑁⟩))

Theoremclwwlkf1 27012* Lemma 3 for clwwlkbij 27015: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkfo 27013* Lemma 4 for clwwlkbij 27015: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkf1o 27014* Lemma 5 for clwwlkbij 27015: F is a 1-1 onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkbij 27015* There is a bijection between the set of closed walks of a fixed length represented by walks (as word) and the set of closed walks (as words) of a fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.)
(𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)}–1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlknwwlkncl 27016* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 22-Mar-2022.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)})

TheoremclwwlknwwlknclOLD 27017* Obsolete version of clwwlknwwlkncl 27016 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ( lastS ‘𝑤) = (𝑤‘0)})

Theoremclwwlkwwlksb 27018 A nonempty word over vertices represents a closed walk iff the word concatenated with its first symbol represents a walk. (Contributed by AV, 4-Mar-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑊 ∈ Word 𝑉𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (WWalks‘𝐺)))

Theoremclwwlknwwlksnb 27019 A word over vertices represents a closed walk of a fixed length 𝑁 greater than zero iff the word concatenated with its first symbol represents a walk of length 𝑁. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, because (𝑊 ++ ⟨“(𝑊‘0)”⟩) = ⟨“∅”⟩ ∈ (0 WWalksN 𝐺) could be true, but not 𝑊 ∈ (0 ClWWalksN 𝐺) ↔ ∅ ∈ ∅. (Contributed by AV, 4-Mar-2022.) (Proof shortened by AV, 22-Mar-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))

Theoremclwwlkext2edg 27020 If a word concatenated with a vertex represents a closed walk in (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑊 ∈ Word 𝑉𝑍𝑉𝑁 ∈ (ℤ‘2)) ∧ (𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑁 ClWWalksN 𝐺)) → ({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸))

Theoremwwlksext2clwwlk 27021 If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Revised by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍𝑉) → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))

Theoremwwlksext2clwwlkOLD 27022 Obsolete version of wwlksext2clwwlk 27021 as of 14-Mar-2022. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 7-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍𝑉𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))

Theoremwwlksubclwwlk 27023 Any prefix of a word representing a closed walk represents a walk. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 28-Apr-2021.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑋 ∈ (𝑁 ClWWalksN 𝐺) → (𝑋 substr ⟨0, 𝑀⟩) ∈ ((𝑀 − 1) WWalksN 𝐺)))

Theoremclwwnisshclwwsn 27024 Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.) (Revised by AV, 29-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))

Theoremeleclclwwlknlem1 27025* Lemma 1 for eleclclwwlkn 27040. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)       ((𝐾 ∈ (0...𝑁) ∧ (𝑋𝑊𝑌𝑊)) → ((𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))

Theoremeleclclwwlknlem2 27026* Lemma 2 for eleclclwwlkn 27040. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)       (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))

Theoremclwwlknscsh 27027* The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 30-Apr-2021.)
((𝑁 ∈ ℕ0𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Theoremumgr2cwwk2dif 27028 If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)
((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘1) ≠ (𝑊‘0))

Theoremumgr2cwwkdifex 27029* If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)
((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^𝑁)(𝑊𝑖) ≠ (𝑊‘0))

Theoremerclwwlknrel 27030 is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       Rel

Theoremerclwwlkneq 27031* Two classes are equivalent regarding if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

Theoremerclwwlkneqlen 27032* If two classes are equivalent regarding , then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 → (#‘𝑇) = (#‘𝑈)))

Theoremerclwwlknref 27033* is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝑥𝑊𝑥 𝑥)

Theoremerclwwlknsym 27034* is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝑥 𝑦𝑦 𝑥)

Theoremerclwwlkntr 27035* is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)

Theoremerclwwlkn 27036* is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}        Er 𝑊

Theoremqerclwwlknfi 27037* The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation is finite. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((Vtx‘𝐺) ∈ Fin → (𝑊 / ) ∈ Fin)

Theoremhashclwwlkn0 27038* The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to . (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((Vtx‘𝐺) ∈ Fin → (#‘𝑊) = Σ𝑥 ∈ (𝑊 / )(#‘𝑥))

Theoremeclclwwlkn1 27039* An equivalence class according to . (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))

Theoremeleclclwwlkn 27040* A member of an equivalence class according to . (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 1-May-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝐵 ∈ (𝑊 / ) ∧ 𝑋𝐵) → (𝑌𝐵 ↔ (𝑌𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

Theoremhashecclwwlkn1 27041* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number is 1 or equals this length. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / )) → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))

Theoremumgrhashecclwwlk 27042* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ) → (#‘𝑈) = 𝑁))

Theoremfusgrhashclwwlkn 27043* The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 1-May-2021.)
𝑊 = (𝑁 ClWWalksN 𝐺)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘𝑊) = ((#‘(𝑊 / )) · 𝑁))

Theoremclwwlkndivn 27044 The size of the set of closed walks (defined as words) of length 𝑁 is divisible by 𝑁 if 𝑁 is a prime number. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 2-May-2021.)
((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘(𝑁 ClWWalksN 𝐺)))

Theoremclwlksfclwwlk2wrd 27045* The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶𝐵 ∈ Word (Vtx‘𝐺))

Theoremclwlksfclwwlk1hashn 27046* The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 2-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)

Theoremclwlksfclwwlk1hash 27047* The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))

Theoremclwlksfclwwlk2sswd 27048* The size of a subword of the second component of a closed walk with length of the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))

Theoremclwlksfclwwlk 27049* There is a function between the set of closed walks (defined as words) of length n and the set of closed walks of length n. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof shortened by AV, 23-Mar-2022.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶(𝑁 ClWWalksN 𝐺))

Theoremclwlksfoclwwlk 27050* There is an onto function between the set of closed walks (defined as words) of a fixed prime length 𝑁 and the set of closed walks of length 𝑁. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof shortened by AV, 23-Mar-2022.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→(𝑁 ClWWalksN 𝐺))

Theoremclwlksf1clwwlklem0 27051* Lemma 1 for clwlksf1clwwlklem 27055. (Contributed by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) ∧ (#‘(1st𝑊)) = 𝑁))

Theoremclwlksf1clwwlklem1 27052* Lemma 1 for clwlksf1clwwlklem 27055. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))

Theoremclwlksf1clwwlklem2 27053* Lemma 2 for clwlksf1clwwlklem 27055. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))

Theoremclwlksf1clwwlklem3 27054* Lemma 3 for clwlksf1clwwlklem 27055. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (2nd𝑊) ∈ Word (Vtx‘𝐺))

Theoremclwlksf1clwwlklem 27055* Lemma for clwlksf1clwwlk 27056. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))

Theoremclwlksf1clwwlk 27056* There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1→(𝑁 ClWWalksN 𝐺))

Theoremclwlksf1oclwwlk 27057* There is a one-to-one onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwlkssizeeq 27058* The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.)
((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (#‘(𝑁 ClWWalksN 𝐺)) = (#‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}))

Theoremclwlksndivn 27059* The size of the set of closed walks of length n is divisible by n. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p". (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.)
((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘(1st𝑐)) = 𝑁}))

16.3.10.3  Closed walks on a vertex of a fixed length as words

Syntaxcclwwlknon 27060 Extend class notation with closed walks (in an undirected graph) anchored at a fixed vertex and of a fixed length as word over the set of vertices.
class ClWWalksNOn

Definitiondf-clwwlknon 27061* Define the set of all closed walks a graph 𝑔, anchored at a fixed vertex 𝑣 (i.e., a walk starting and ending at the fixed vertex 𝑣, also called "a closed walk on vertex 𝑣") and having a fixed length 𝑛 as words over the set of vertices. Such a word corresponds to the sequence v=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)=v as defined in df-clwlks 26723. The set ((𝑣(ClWWalksNOn‘𝑔)𝑛) corresponds to the set of "walks from v to v of length n" in a statement of [Huneke] p. 2. (Contributed by AV, 24-Feb-2022.)
ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))

Theoremclwwlknonmpt2 27062* (ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.)
(ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})

Theoremclwwlknon 27063* The set of closed walks on vertex 𝑋 of length 𝑁 in a graph 𝐺 as words over the set of vertices. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)
(𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}

TheoremclwwlknonOLD 27064* Obsolete version of clwwlknon 27063 as of 24-Mar-2022. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 25-Feb-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Theoremisclwwlknon 27065 A word over the set of vertices representing a closed walk on vertex 𝑋 of length 𝑁 in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Revised by AV, 24-Mar-2022.)
(𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))

TheoremisclwwlknonOLD 27066 Obsolete version of isclwwlknon 27065 as of 24-Mar-2022. (Contributed by AV, 25-Feb-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋)))

Theoremclwwlk0on0 27067 There is no word over the set of vertices representing a closed walk on vertex 𝑋 of length 0 in a graph 𝐺. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.)
(𝑋(ClWWalksNOn‘𝐺)0) = ∅

Theoremclwwlknon0 27068 Sufficient conditions for ClWWalksNOn to be empty. (Contributed by AV, 25-Mar-2022.)
(¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)

Theoremclwwlknonfin 27069 In a finite graph 𝐺, the set of closed walks on vertex 𝑋 of length 𝑁 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin)

Theoremclwwlknonel 27070* Characterization of a word over the set of vertices representing a closed walk on vertex 𝑋 of (nonzero) length 𝑁 in a graph 𝐺. This theorem would not hold for 𝑁 = 0 if 𝑊 = 𝑋 = ∅. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ≠ 0 → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋)))

TheoremclwwlknonelOLD 27071* Obsolete version of clwwlknonel 27070 as of 24-Mar-2022. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 25-Feb-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋)))

Theoremclwwlknon1 27072* The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})

Theoremclwwlknon1loop 27073 If there is a loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑋𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})

Theoremclwwlknon1nloop 27074 If there is no loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is empty. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅)

Theoremclwwlknon1sn 27075 The set of (closed) walks on vertex 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋 iff there is a loop at 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (ClWWalksNOn‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑋𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑋} ∈ 𝐸))

Theoremclwwlknon1le1 27076 There is at most one (closed) walk on vertex 𝑋 of length 1 as word over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
(#‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1

Theoremclwwlknon2 27077* The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 5-Mar-2022.) (Proof revised by AV, 25-Mar-2022.)
𝐶 = (ClWWalksNOn‘𝐺)       (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}

Theoremclwwlknon2x 27078* The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.)
𝐶 = (ClWWalksNOn‘𝐺)    &   𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}

Theoremclwwlknon2num 27079 In a 𝐾-regular graph 𝐺, there are 𝐾 closed walks on vertex 𝑋 of length 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)
((𝐺RegUSGraph𝐾𝑋 ∈ (Vtx‘𝐺)) → (#‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)

Theoremclwwlknonwwlknonb 27080 A word over vertices represents a closed walk of a fixed length 𝑁 on vertex 𝑋 iff the word concatenated with 𝑋 represents a walk of length 𝑁 on 𝑋 and 𝑋. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, see clwwlknwwlksnb 27019. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

TheoremclwwlknonwwlknonbOLD 27081 Obsolete version of clwwlknonwwlknonb 27080 as of 27-Mar-2022. (Contributed by AV, 4-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝑊 ∈ Word 𝑉𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))

Theoremclwwlknonex2lem1 27082 Lemma 1 for clwwlknonex2 27084: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. This Lemma would not hold for 𝑁 = 2, i.e., (#‘𝑊) = 0, because (0..^(((#‘𝑊) + 2) − 1)) = (0..^((0 + 2) − 1)) = (0..^1) = {0} ≠ {-1, 0} = (∅ ∪ {-1, 0}) = ((0..^(0 − 1)) ∪ {(0 − 1), 0}) = ((0..^((#‘𝑊) − 1)) ∪ {((#‘𝑊) − 1), (#‘𝑊)}). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.)
((𝑁 ∈ (ℤ‘3) ∧ (#‘𝑊) = (𝑁 − 2)) → (0..^(((#‘𝑊) + 2) − 1)) = ((0..^((#‘𝑊) − 1)) ∪ {((#‘𝑊) − 1), (#‘𝑊)}))

Theoremclwwlknonex2lem2 27083* Lemma 2 for clwwlknonex2 27084: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋)) ∧ {𝑋, 𝑌} ∈ 𝐸) → ∀𝑖 ∈ ((0..^((#‘𝑊) − 1)) ∪ {((#‘𝑊) − 1), (#‘𝑊)}){(((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑖), (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑖 + 1))} ∈ 𝐸)

Theoremclwwlknonex2 27084 Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑁 ClWWalksN 𝐺))

Theoremclwwlknonex2e 27085 Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk on vertex 𝑋. (Contributed by AV, 17-Apr-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑋𝑉𝑌𝑉𝑁 ∈ (ℤ‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Theoremclwwlknondisj 27086* The sets of closed walks on different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
Disj 𝑥𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)

Theoremclwwlknun 27087* The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = 𝑥𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))

Theoremclwwlkvbij 27088* There is a bijection between the set of closed walks of a fixed length 𝑁 on a fixed vertex 𝑋 represented by walks (as word) and the set of closed walks (as words) of the fixed length 𝑁 on the fixed vertex 𝑋. The difference between these two representations is that in the first case the fixed vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑋𝑉𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

TheoremclwwlkvbijOLD 27089* Obsolete version of clwwlkvbij 27088 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑆})

TheoremclwwlknondisjOLD 27090* Obsolete version of clwwlknondisj 27086 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Disj 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥}

TheoremclwwlknunOLD 27091* Obsolete version of clwwlknun 27087 as of 3-Mar-2022. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0) → (𝑁 ClWWalksN 𝐺) = 𝑥𝑉 {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑥})

16.3.11  Examples for walks, trails and paths

Theorem0ewlk 27092 The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.)
((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ∅ ∈ (𝐺 EdgWalks 𝑆))

Theorem1ewlk 27093 A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)
((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))

Theorem0wlk 27094 A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑈 → (∅(Walks‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Theoremis0wlk 27095 A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁𝑉) → ∅(Walks‘𝐺)𝑃)

Theorem0wlkonlem1 27096 Lemma 1 for 0wlkon 27098 and 0trlon 27102. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → (𝑁𝑉𝑁𝑉))

Theorem0wlkonlem2 27097 Lemma 2 for 0wlkon 27098 and 0trlon 27102. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉pm (0...0)))

Theorem0wlkon 27098 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)𝑃)

Theorem0wlkons1 27099 A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)⟨“𝑁”⟩)

Theorem0trl 27100 A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑈 → (∅(Trails‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

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