P. 301 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clwwlksclwwlkn 30001	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)
Theorem	clwwlknlen 30002	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)
Theorem	clwwlknnn 30003	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)
Theorem	clwwlknwrd 30004	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)
Theorem	clwwlknbp 30005	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))
Theorem	isclwwlknx 30006*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁)))
Theorem	clwwlknp 30007*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))
Theorem	clwwlknwwlksn 30008	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 𝑎 ≠ 𝑐) 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))
Theorem	clwwlknlbonbgr1 30009	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)))
Theorem	clwwlkinwwlk 30010	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 ∈ (ℤ≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
Theorem	clwwlkn1 30011	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)))
Theorem	loopclwwlkn1b 30012	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)))
Theorem	clwwlkn1loopb 30013*	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)))
Theorem	clwwlkn2 30014	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)))
Theorem	clwwlknfi 30015	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)
Theorem	clwwlkel 30016*	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)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ 𝐷)
Theorem	clwwlkf 30017*	Lemma 1 for clwwlkf1o 30021: 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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkfv 30018*	Lemma 2 for clwwlkf1o 30021: 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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))
Theorem	clwwlkf1 30019*	Lemma 3 for clwwlkf1o 30021: 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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkfo 30020*	Lemma 4 for clwwlkf1o 30021: 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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkf1o 30021*	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 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1-onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlken 30022*	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))
Theorem	clwwlknwwlkncl 30023*	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)})
Theorem	clwwlkwwlksb 30024	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 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (WWalks‘𝐺)))
Theorem	clwwlknwwlksnb 30025	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 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
Theorem	clwwlkext2edg 30026	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 𝑉 ∧ 𝑍 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ (𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑁 ClWWalksN 𝐺)) → ({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸))
Theorem	wwlksext2clwwlk 30027	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) ∧ 𝑍 ∈ 𝑉) → (({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
Theorem	wwlksubclwwlk 30028	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)))
Theorem	clwwnisshclwwsn 30029	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))
Theorem	eleclclwwlknlem1 30030*	Lemma 1 for eleclclwwlkn 30046. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    ⇒   ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊)) → ((𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
Theorem	eleclclwwlknlem2 30031*	Lemma 2 for eleclclwwlkn 30046. (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)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
Theorem	clwwlknscsh 30032*	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 ∈ ℕ0 ∧ 𝑊 ∈ (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)})
Theorem	clwwlknccat 30033	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))
Theorem	umgr2cwwk2dif 30034	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))
Theorem	umgr2cwwkdifex 30035*	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)(𝑊‘𝑖) ≠ (𝑊‘0))
Theorem	erclwwlknrel 30036	∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ Rel ∼
Theorem	erclwwlkneq 30037*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
Theorem	erclwwlkneqlen 30038*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (♯‘𝑇) = (♯‘𝑈)))
Theorem	erclwwlknref 30039*	∼ 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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥)
Theorem	erclwwlknsym 30040*	∼ 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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)
Theorem	erclwwlkntr 30041*	∼ 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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)
Theorem	erclwwlkn 30042*	∼ 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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ∼ Er 𝑊
Theorem	qerclwwlknfi 30043*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ∈ Fin)
Theorem	hashclwwlkn0 30044*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(♯‘𝑥))
Theorem	eclclwwlkn1 30045*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
Theorem	eleclclwwlkn 30046*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
Theorem	hashecclwwlkn1 30047*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))
Theorem	umgrhashecclwwlk 30048*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (♯‘𝑈) = 𝑁))
Theorem	fusgrhashclwwlkn 30049*	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⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑊) = ((♯‘(𝑊 / ∼ )) · 𝑁))
Theorem	clwwlkndivn 30050	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)))
Theorem	clwlknf1oclwwlknlem1 30051	Lemma 1 for clwlknf1oclwwlkn 30054. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))
Theorem	clwlknf1oclwwlknlem2 30052*	Lemma 2 for clwlknf1oclwwlkn 30054: 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) ∣ (♯‘(1st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st ‘𝑐)) ∧ (♯‘(1st ‘𝑐)) = 𝑁)})
Theorem	clwlknf1oclwwlknlem3 30053*	Lemma 3 for clwlknf1oclwwlkn 30054: The bijective function of clwlknf1oclwwlkn 30054 is the bijective function of clwlkclwwlkf1o 29981 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐴 = (1st ‘𝑐)    &   ⊢ 𝐵 = (2nd ‘𝑐)    &   ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
Theorem	clwlknf1oclwwlkn 30054*	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 = (1st ‘𝑐)    &   ⊢ 𝐵 = (2nd ‘𝑐)    &   ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwlkssizeeq 30055*	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) ∣ (♯‘(1st ‘𝑤)) = 𝑁}))
Theorem	clwlksndivn 30056*	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) ∣ (♯‘(1st ‘𝑐)) = 𝑁}))
17.3.10.3  Closed walks on a vertex of a fixed length as words
Syntax	cclwwlknon 30057	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
Definition	df-clwwlknon 30058*	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 29742. 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}))
Theorem	clwwlknonmpo 30059*	(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.) (Proof shortened by AV, 2-Mar-2024.)
⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
Theorem	clwwlknon 30060*	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}
Theorem	isclwwlknon 30061	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))
Theorem	clwwlk0on0 30062	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) = ∅
Theorem	clwwlknon0 30063	Sufficient conditions for ClWWalksNOn to be empty. (Contributed by AV, 25-Mar-2022.)
⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)
Theorem	clwwlknonfin 30064	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)
Theorem	clwwlknonel 30065*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋)))
Theorem	clwwlknonccat 30066	The concatenation of two words representing closed walks on a vertex 𝑋 represents a closed walk on vertex 𝑋. The resulting walk is a "double loop", starting at vertex 𝑋, coming back to 𝑋 by the first walk, following the second walk and finally coming back to 𝑋 again. (Contributed by AV, 24-Apr-2022.)
⊢ ((𝐴 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑀) ∧ 𝐵 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) → (𝐴 ++ 𝐵) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑀 + 𝑁)))
Theorem	clwwlknon1 30067*	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)    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Theorem	clwwlknon1loop 30068	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)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})
Theorem	clwwlknon1nloop 30069	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) = ∅)
Theorem	clwwlknon1sn 30070	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)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑋} ∈ 𝐸))
Theorem	clwwlknon1le1 30071	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
Theorem	clwwlknon2 30072*	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.) (Revised by AV, 25-Mar-2022.)
⊢ 𝐶 = (ClWWalksNOn‘𝐺)    ⇒   ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
Theorem	clwwlknon2x 30073*	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)}
Theorem	s2elclwwlknon2 30074	Sufficient conditions of a doubleton word to represent a closed walk on vertex 𝑋 of length 2. (Contributed by AV, 11-May-2022.)
⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → ⟨“𝑋𝑌”⟩ ∈ (𝑋𝐶2))
Theorem	clwwlknon2num 30075	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.)
⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
Theorem	clwwlknonwwlknonb 30076	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 30025. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))
Theorem	clwwlknonex2lem1 30077	Lemma 1 for clwwlknonex2 30079: 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)}))
Theorem	clwwlknonex2lem2 30078*	Lemma 2 for clwwlknonex2 30079: 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)    ⇒   ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (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)
Theorem	clwwlknonex2 30079	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)    ⇒   ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑁 ClWWalksN 𝐺))
Theorem	clwwlknonex2e 30080	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)    ⇒   ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
Theorem	clwwlknondisj 30081*	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 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)
Theorem	clwwlknun 30082*	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) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))
Theorem	clwwlkvbij 30083*	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, 7-Jul-2022.) (Proof shortened by AV, 2-Nov-2022.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
17.3.11  Examples for walks, trails and paths
Theorem	0ewlk 30084	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))
Theorem	1ewlk 30085	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))
Theorem	0wlk 30086	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)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	is0wlk 30087	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⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Walks‘𝐺)𝑃)
Theorem	0wlkonlem1 30088	Lemma 1 for 0wlkon 30090 and 0trlon 30094. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → (𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
Theorem	0wlkonlem2 30089	Lemma 2 for 0wlkon 30090 and 0trlon 30094. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉 ↑pm (0...0)))
Theorem	0wlkon 30090	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)
Theorem	0wlkons1 30091	A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)⟨“𝑁”⟩)
Theorem	0trl 30092	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)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	is0trl 30093	A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Trails‘𝐺)𝑃)
Theorem	0trlon 30094	A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-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(TrailsOn‘𝐺)𝑁)𝑃)
Theorem	0pth 30095	A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	0spth 30096	A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	0pthon 30097	A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(PathsOn‘𝐺)𝑁)𝑃)
Theorem	0pthon1 30098	A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){⟨0, 𝑁⟩})
Theorem	0pthonv 30099*	For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∃𝑓∃𝑝 𝑓(𝑁(PathsOn‘𝐺)𝑁)𝑝)
Theorem	0clwlk 30100	A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑋 → (∅(ClWalks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))