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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremclwlkclwwlkfo 27401* 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 27402* 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 27403* 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 27404* 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))

TheoremclwlkclwwlkenOLD 27405* Obsolete proof of clwlkclwwlken 27404 as of 12-Oct-2022. (Contributed by AV, 25-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ USPGraph → {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ≈ (ClWWalks‘𝐺))

Theoremclwwisshclwwslemlem 27406* Lemma for clwwisshclwwslem 27407. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
(((𝐿 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅)

Theoremclwwisshclwwslem 27407* Lemma for clwwisshclwws 27408. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(♯‘𝑊))) → ((∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) → ∀𝑗 ∈ (0..^((♯‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ 𝐸))

Theoremclwwisshclwws 27408 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)
((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺))

Theoremclwwisshclwwsn 27409 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jun-2018.) (Revised by AV, 29-Apr-2021.)
((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(♯‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺))

Theoremerclwwlkrel 27410 is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       Rel

Theoremerclwwlkeq 27411* Two classes are equivalent regarding if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))

Theoremerclwwlkeqlen 27412* 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, 29-Apr-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 → (♯‘𝑈) = (♯‘𝑊)))

Theoremerclwwlkref 27413* is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 𝑥)

Theoremerclwwlksym 27414* is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       (𝑥 𝑦𝑦 𝑥)

Theoremerclwwlktr 27415* 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.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)

Theoremerclwwlk 27416* is an equivalence 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.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}        Er (ClWWalks‘𝐺)

16.3.10.2  Closed walks of a fixed length as words

Syntaxcclwwlkn 27417 Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices.
class ClWWalksN

Definitiondf-clwwlkn 27418* Define the set of all closed walks of a fixed length 𝑛 as words over the set of vertices in a graph 𝑔. If 0 < 𝑛, 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 27127. For 𝑛 = 0, the set is empty, see clwwlkn0 27421. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)
ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})

Theoremclwwlkn 27419* The set of closed walks of a fixed length 𝑁 as words over the set of vertices in a graph 𝐺. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2021.)
(𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}

Theoremisclwwlkn 27420 A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2021.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))

Theoremclwwlkn0 27421 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) = ∅

Theoremclwwlkneq0 27422 Sufficient conditions for ClWWalksN to be empty. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 24-Feb-2022.)
((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅)

Theoremclwwlkclwwlkn 27423 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.) (Proof shortened by AV, 22-Mar-2022.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ (ClWWalks‘𝐺))

Theoremclwwlksclwwlkn 27424 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)

Theoremclwwlknlen 27425 The length of a word representing a closed walk of a fixed length is this fixed length. (Contributed by AV, 22-Mar-2022.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (♯‘𝑊) = 𝑁)

Theoremclwwlknnn 27426 The length of a closed walk of a fixed length as word is a positive integer. (Contributed by AV, 22-Mar-2022.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 ∈ ℕ)

Theoremclwwlknwrd 27427 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)

Theoremclwwlknbp 27428 Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁))

Theoremisclwwlknx 27429* Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
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)))

Theoremclwwlknp 27430* 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.) (Proof shortened by AV, 23-Mar-2022.)
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))

Theoremclwwlknwwlksn 27431 A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 − 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if ⟨“𝑎𝑏𝑐”⟩ ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then ⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if 𝑎𝑐) of length 2, and ⟨“𝑎𝑏𝑐𝑎”⟩ ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) (Revised by AV, 22-Mar-2022.)
(𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺))

Theoremclwwlknlbonbgr1 27432 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 27433 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.) (Revised by AV, 30-Oct-2022.)
(((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))

TheoremclwwlkinwwlkOLD 27434 Obsolete version of clwwlkinwwlk 27433 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Proof shortened by AV, 23-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊𝑁) = (𝑊‘0)) → (𝑊 substr ⟨0, 𝑁⟩) ∈ (𝑁 ClWWalksN 𝐺))

Theoremclwwlkn1 27435 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 27436 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 27437* 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 27438 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 27439 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.) (Proof shortened by JJ, 18-Nov-2022.)
((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)

TheoremclwwlknfiOLD 27440 Obsolete version of clwwlknfi 27439 as of 4-May-2023. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)

Theoremclwwlkel 27441* 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)

TheoremclwwlkfOLD 27442* Obsolete version of clwwlkf 27447 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))

TheoremclwwlkfvOLD 27443* Obsolete version of clwwlkfv 27448 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑊𝐷 → (𝐹𝑊) = (𝑊 substr ⟨0, 𝑁⟩))

Theoremclwwlkf1OLD 27444* Obsolete version of clwwlkf1 27449 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))

TheoremclwwlkfoOLD 27445* Obsolete version of clwwlkfo 27450 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkf1oOLD 27446* Obsolete version of clwwlkf1o 27451 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkf 27447* Lemma 1 for clwwlkf1o 27451: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))       (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))

Theoremclwwlkfv 27448* Lemma 2 for clwwlkf1o 27451: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))       (𝑊𝐷 → (𝐹𝑊) = (𝑊 prefix 𝑁))

Theoremclwwlkf1 27449* Lemma 3 for clwwlkf1o 27451: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkfo 27450* Lemma 4 for clwwlkf1o 27451: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))       (𝑁 ∈ ℕ → 𝐹:𝐷onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlkf1o 27451* F is a 1-1 onto function, that means that there is a bijection between the set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the 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.) (Revised by AV, 1-Nov-2022.)
𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 prefix 𝑁))       (𝑁 ∈ ℕ → 𝐹:𝐷1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwwlken 27452* The set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length are equinumerous. (Contributed by AV, 5-Jun-2022.) (Proof shortened by AV, 2-Nov-2022.)
(𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))

TheoremclwwlkenOLD 27453* Obsolete proof of clwwlken 27452 as of 12-Oct-2022. (Contributed by AV, 5-Jun-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))

Theoremclwwlknwwlkncl 27454* 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)})

Theoremclwwlkwwlksb 27455 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 27456 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 27457 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 27458 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)))

Theoremwwlksubclwwlk 27459 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.) (Revised by AV, 1-Nov-2022.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑋 ∈ (𝑁 ClWWalksN 𝐺) → (𝑋 prefix 𝑀) ∈ ((𝑀 − 1) WWalksN 𝐺)))

TheoremwwlksubclwwlkOLD 27460 Obsolete version of wwlksubclwwlk 27459 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 28-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑋 ∈ (𝑁 ClWWalksN 𝐺) → (𝑋 substr ⟨0, 𝑀⟩) ∈ ((𝑀 − 1) WWalksN 𝐺)))

Theoremclwwnisshclwwsn 27461 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 27462* Lemma 1 for eleclclwwlkn 27478. (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 27463* Lemma 2 for eleclclwwlkn 27478. (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 27464* 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)})

Theoremclwwlknccat 27465 The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. 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, 24-Apr-2022.)
((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐵 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺))

Theoremumgr2cwwk2dif 27466 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 27467* 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 27468 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 27469* 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 27470* 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 27471* 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 27472* 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 27473* 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 27474* 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 27475* 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 27476* 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 27477* 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 27478* 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 27479* 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 27480* 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 27481* 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 27482 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)))

Theoremclwlknf1oclwwlknlem1 27483 Lemma 1 for clwlknf1oclwwlkn 27487. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) prefix ((♯‘(2nd𝐶)) − 1))) = (♯‘(1st𝐶)))

Theoremclwlknf1oclwwlknlem1OLD 27484 Obsolete version of clwlknf1oclwwlknlem1 27483 as of 12-Oct-2022. (Contributed by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st𝐶))) → (♯‘((2nd𝐶) substr ⟨0, ((♯‘(2nd𝐶)) − 1)⟩)) = (♯‘(1st𝐶)))

Theoremclwlknf1oclwwlknlem2 27485* Lemma 2 for clwlknf1oclwwlkn 27487: The closed walks of a positive length are nonempty closed walks of this length. (Contributed by AV, 26-May-2022.)
(𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st𝑐)) ∧ (♯‘(1st𝑐)) = 𝑁)})

Theoremclwlknf1oclwwlknlem3 27486* Lemma 3 for clwlknf1oclwwlkn 27487: The bijective function of clwlknf1oclwwlkn 27487 is the bijective function of clwlkclwwlkf1o 27403 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))

Theoremclwlknf1oclwwlkn 27487* There is a one-to-one onto function between the set of closed walks as words of length 𝑁 and the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 1-Nov-2022.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 prefix (♯‘𝐴)))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwlknf1oclwwlknlem3OLD 27488* Obsolete version of clwlknf1oclwwlknlem3 27486 as of 12-Oct-2022. (Contributed by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))} ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩)) ↾ 𝐶))

Theoremclwlknf1oclwwlknOLD 27489* Obsolete version of clwlknf1oclwwlkn 27487 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶1-1-onto→(𝑁 ClWWalksN 𝐺))

Theoremclwlkssizeeq 27490* The size of the set of closed walks as words of length 𝑁 corresponds to the size of the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Revised by AV, 26-May-2022.) (Proof shortened by AV, 3-Nov-2022.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}))

TheoremclwlkssizeeqOLD 27491* Obsolete proof of clwlkssizeeq 27490 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Revised by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st𝑤)) = 𝑁}))

Theoremclwlksndivn 27492* The size of the set of closed walks of prime 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 27493 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 27494* 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 27127. 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 27495* (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 27496* 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}

Theoremisclwwlknon 27497 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))

Theoremclwwlk0on0 27498 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 27499 Sufficient conditions for ClWWalksNOn to be empty. (Contributed by AV, 25-Mar-2022.)
(¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)

Theoremclwwlknonfin 27500 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)

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