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
17.3.9  Walks in regular graphs
Theorem	rusgrnumwwlkl1 30001*	In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Theorem	rusgrnumwwlkslem 30002*	Lemma for rusgrnumwwlks 30007. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)})
Theorem	rusgrnumwwlklem 30003*	Lemma for rusgrnumwwlk 30008 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
Theorem	rusgrnumwwlkb0 30004*	Induction base 0 for rusgrnumwwlk 30008. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
Theorem	rusgrnumwwlkb1 30005*	Induction base 1 for rusgrnumwwlk 30008. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿1) = 𝐾)
Theorem	rusgr0edg 30006*	Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 RegUSGraph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
Theorem	rusgrnumwwlks 30007*	Induction step for rusgrnumwwlk 30008. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 27-May-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Theorem	rusgrnumwwlk 30008*	In a 𝐾-regular graph, the number of walks of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. By definition, (𝑁 WWalksN 𝐺) is the set of walks (as words) with length 𝑁, and (𝑃𝐿𝑁) is the number of walks with length 𝑁 starting at the vertex 𝑃. Because of the 𝐾-regularity, a walk can be continued in 𝐾 different ways at the end vertex of the walk, and this repeated 𝑁 times. This theorem even holds for 𝑁 = 0: in this case, the walk consists of only one vertex 𝑃, so the number of walks of length 𝑁 = 0 starting with 𝑃 is (𝐾↑0) = 1. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (♯‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))
Theorem	rusgrnumwwlkg 30009*	In a 𝐾-regular graph, the number of walks (as words) of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. Closed form of rusgrnumwwlk 30008. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁))
Theorem	rusgrnumwlkg 30010*	In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular." This theorem even holds for n=0: then the walk consists of only one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.) (Proof shortened by AV, 5-Aug-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘{𝑤 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁))
Theorem	clwwlknclwwlkdif 30011*	The set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑉 and ending not at this vertex is the difference between the set 𝐶 of walks of length 𝑁 starting with this vertex 𝑋 and the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)    &   ⊢ 𝐶 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}    ⇒   ⊢ 𝐴 = (𝐶 ∖ 𝐵)
Theorem	clwwlknclwwlkdifnum 30012*	In a 𝐾-regular graph, the size of the set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 8-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (♯‘𝐴) = ((𝐾↑𝑁) − (♯‘𝐵)))
17.3.10  Closed walks as words In general, a closed walk is an alternating sequence of vertices and edges, as defined in df-clwlks 29807: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n), with p(n) = p(0). Often, it is sufficient to refer to a walk by the (cyclic) sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(0), see the corresponding remark on cycles (which are special closed walks) in [Diestel] p. 7. As for "walks as words" in general, the concept of a Word, see df-word 14563, is also used in Definitions df-clwwlk 30014 and df-clwwlkn 30057, and the representation of a closed walk as the sequence of its vertices is called "closed walk as word". In contrast to "walks as words", the terminating vertex p(n) of a closed walk is omitted in the representation of a closed walk as word, see definitions df-clwwlk 30014, df-clwwlkn 30057 and df-clwwlknon 30120, because it is always equal to the first vertex of the closed walk. This represenation has the advantage that the vertices can be cyclically shifted without changing the represented closed walk. Furthermore, the length of a closed walk (i.e. the number of its edges) equals the number of symbols/vertices of the word representing the closed walk. To avoid to handle the degenerate case of representing a (closed) walk of length 0 by the empty word, this case is excluded within the definition (𝑤 ≠ ∅). This is because a walk of length 0 is anchored at an arbitrary vertex by the general definition for closed walks, see 0clwlkv 30163, which neither can be reflected by the empty word nor by a singleton word ⟨“𝑣”⟩ with vertex v : ⟨“𝑣”⟩ represents the walk "𝑣𝑣", which is a (closed) walk of length 1 (if there is an edge/loop from 𝑣 to 𝑣), see loopclwwlkn1b 30074. Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 30035 or clwlkclwwlken 30044. Although the set ClWWalksN of all closed walks of a fixed length as words over the set of vertices is defined as function over ℕ0, the fixed length is usually not 0, because (0 ClWWalksN 𝐺) = ∅ (see clwwlkn0 30060). Analogous to (𝐴(𝑁 WWalksNOn 𝐺)𝐵), the set of walks of a fixed length 𝑁 between two vertices 𝐴 and 𝐵, the set (𝑋(ClWWalksNOn‘𝐺)𝑁) of closed walks of a fixed length 𝑁 anchored at a fixed vertex 𝑋 is defined by df-clwwlknon 30120. This definition is also based on ℕ0 instead of ℕ, with (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 30124). clwwlknon1le1 30133 states that there is at most one (closed) walk of length 1 on a vertex, which would consist of a loop (see clwwlknon1loop 30130). And in a 𝐾-regular graph, there are 𝐾 closed walks of length 2 on each vertex, see clwwlknon2num 30137.
17.3.10.1  Closed walks as words
Syntax	cclwwlk 30013	Extend class notation with closed walks (in an undirected graph) as word over the set of vertices.
class ClWWalks
Definition	df-clwwlk 30014*	Define the set of all closed walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlks 29807. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})
Theorem	clwwlk 30015*	The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (ClWWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑤), (𝑤‘0)} ∈ 𝐸)}
Theorem	isclwwlk 30016*	Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))
Theorem	clwwlkbp 30017	Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
Theorem	clwwlkgt0 30018	There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)
⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 0 < (♯‘𝑊))
Theorem	clwwlksswrd 30019	Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.)
⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺)
Theorem	clwwlk1loop 30020	A closed walk of length 1 is a loop. See also clwlkl1loop 29819. (Contributed by AV, 24-Apr-2021.)
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))
Theorem	clwwlkccatlem 30021*	Lemma for clwwlkccat 30022: index 𝑗 is shifted up by (♯‘𝐴), and the case 𝑖 = ((♯‘𝐴) − 1) is covered by the "bridge" {(lastS‘𝐴), (𝐵‘0)} = {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺). (Contributed by AV, 23-Apr-2022.)
⊢ ((((𝐴 ∈ Word (Vtx‘𝐺) ∧ 𝐴 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((♯‘𝐴) − 1)){(𝐴‘𝑖), (𝐴‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐴), (𝐴‘0)} ∈ (Edg‘𝐺)) ∧ ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐵 ≠ ∅) ∧ ∀𝑗 ∈ (0..^((♯‘𝐵) − 1)){(𝐵‘𝑗), (𝐵‘(𝑗 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝐵), (𝐵‘0)} ∈ (Edg‘𝐺)) ∧ (𝐴‘0) = (𝐵‘0)) → ∀𝑖 ∈ (0..^((♯‘(𝐴 ++ 𝐵)) − 1)){((𝐴 ++ 𝐵)‘𝑖), ((𝐴 ++ 𝐵)‘(𝑖 + 1))} ∈ (Edg‘𝐺))
Theorem	clwwlkccat 30022	The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 23-Apr-2022.)
⊢ ((𝐴 ∈ (ClWWalks‘𝐺) ∧ 𝐵 ∈ (ClWWalks‘𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ (ClWWalks‘𝐺))
Theorem	umgrclwwlkge2 30023	A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.)
⊢ (𝐺 ∈ UMGraph → (𝑃 ∈ (ClWWalks‘𝐺) → 2 ≤ (♯‘𝑃)))
Theorem	clwlkclwwlklem2a1 30024*	Lemma 1 for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))
Theorem	clwlkclwwlklem2a2 30025*	Lemma 2 for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1))
Theorem	clwlkclwwlklem2a3 30026*	Lemma 3 for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (𝑃‘(♯‘𝐹)) = (lastS‘𝑃))
Theorem	clwlkclwwlklem2fv1 30027*	Lemma 4a for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 2))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))
Theorem	clwlkclwwlklem2fv2 30028*	Lemma 4b for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 2 ≤ (♯‘𝑃)) → (𝐹‘((♯‘𝑃) − 2)) = (◡𝐸‘{(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)}))
Theorem	clwlkclwwlklem2a4 30029*	Lemma 4 for clwlkclwwlklem2a 30030. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → ({(𝑃‘𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹‘𝐼)) = {(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})))
Theorem	clwlkclwwlklem2a 30030*	Lemma for clwlkclwwlklem2 30032. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ if(𝑥 < ((♯‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))
Theorem	clwlkclwwlklem1 30031*	Lemma 1 for clwlkclwwlk 30034. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓)))))
Theorem	clwlkclwwlklem2 30032*	Lemma 2 for clwlkclwwlk 30034. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ 2 ≤ (♯‘𝑃)) ∧ (∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))) → ((lastS‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((♯‘𝐹) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸))
Theorem	clwlkclwwlklem3 30033*	Lemma 3 for clwlkclwwlk 30034. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(♯‘𝑓))) ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((♯‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((♯‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))
Theorem	clwlkclwwlk 30034*	A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 30-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)𝑃 ↔ ((lastS‘𝑃) = (𝑃‘0) ∧ (𝑃 prefix ((♯‘𝑃) − 1)) ∈ (ClWWalks‘𝐺))))
Theorem	clwlkclwwlk2 30035*	A closed walk corresponds to a closed walk as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 2-Nov-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))
Theorem	clwlkclwwlkflem 30036*	Lemma for clwlkclwwlkf 30040. (Contributed by AV, 24-May-2022.)
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐴 = (1st ‘𝑈)    &   ⊢ 𝐵 = (2nd ‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
Theorem	clwlkclwwlkf1lem2 30037*	Lemma 2 for clwlkclwwlkf1 30042. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.)
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐴 = (1st ‘𝑈)    &   ⊢ 𝐵 = (2nd ‘𝑈)    &   ⊢ 𝐷 = (1st ‘𝑊)    &   ⊢ 𝐸 = (2nd ‘𝑊)    ⇒   ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))
Theorem	clwlkclwwlkf1lem3 30038*	Lemma 3 for clwlkclwwlkf1 30042. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 30-Oct-2022.)
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐴 = (1st ‘𝑈)    &   ⊢ 𝐵 = (2nd ‘𝑈)    &   ⊢ 𝐷 = (1st ‘𝑊)    &   ⊢ 𝐸 = (2nd ‘𝑊)    ⇒   ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ∀𝑖 ∈ (0...(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))
Theorem	clwlkclwwlkfolem 30039*	Lemma for clwlkclwwlkfo 30041. (Contributed by AV, 25-May-2022.)
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}    ⇒   ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)
Theorem	clwlkclwwlkf 30040*	𝐹 is a function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by AV, 23-May-2022.) (Revised by AV, 29-Oct-2022.)
⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))
Theorem	clwlkclwwlkfo 30041*	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 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–onto→(ClWWalks‘𝐺))
Theorem	clwlkclwwlkf1 30042*	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 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1→(ClWWalks‘𝐺))
Theorem	clwlkclwwlkf1o 30043*	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 ≤ (♯‘(1st ‘𝑤))}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2nd ‘𝑐) prefix ((♯‘(2nd ‘𝑐)) − 1)))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶–1-1-onto→(ClWWalks‘𝐺))
Theorem	clwlkclwwlken 30044*	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 ≤ (♯‘(1st ‘𝑤))} ≈ (ClWWalks‘𝐺))
Theorem	clwwisshclwwslemlem 30045*	Lemma for clwwisshclwwslem 30046. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
⊢ (((𝐿 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅)
Theorem	clwwisshclwwslem 30046*	Lemma for clwwisshclwws 30047. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(♯‘𝑊))) → ((∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (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))
Theorem	clwwisshclwws 30047	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))
Theorem	clwwisshclwwsn 30048	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))
Theorem	erclwwlkrel 30049	∼ 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 ∼
Theorem	erclwwlkeq 30050*	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))}    ⇒   ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(♯‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
Theorem	erclwwlkeqlen 30051*	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))}    ⇒   ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (♯‘𝑈) = (♯‘𝑊)))
Theorem	erclwwlkref 30052*	∼ 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)
Theorem	erclwwlksym 30053*	∼ 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 ∼ 𝑦 → 𝑦 ∼ 𝑥)
Theorem	erclwwlktr 30054*	∼ 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 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)
Theorem	erclwwlk 30055*	∼ 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)
17.3.10.2  Closed walks of a fixed length as words
Syntax	cclwwlkn 30056	Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices.
class ClWWalksN
Definition	df-clwwlkn 30057*	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 29807. For 𝑛 = 0, the set is empty, see clwwlkn0 30060. (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})
Theorem	clwwlkn 30058*	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-2022.)
⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (♯‘𝑤) = 𝑁}
Theorem	isclwwlkn 30059	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-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 𝑁))
Theorem	clwwlkn0 30060	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) = ∅
Theorem	clwwlkneq0 30061	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) = ∅)
Theorem	clwwlkclwwlkn 30062	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))
Theorem	clwwlksclwwlkn 30063	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 30064	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 30065	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 30066	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 30067	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 30068*	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 30069*	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 30070	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 30071	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 30072	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 30073	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 30074	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 30075*	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 30076	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 30077	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 30078*	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 30079*	Lemma 1 for clwwlkf1o 30083: 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 30080*	Lemma 2 for clwwlkf1o 30083: 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 30081*	Lemma 3 for clwwlkf1o 30083: 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 30082*	Lemma 4 for clwwlkf1o 30083: 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 30083*	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 30084*	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 30085*	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 30086	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 30087	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 30088	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 30089	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 30090	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 30091	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 30092*	Lemma 1 for eleclclwwlkn 30108. (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 30093*	Lemma 2 for eleclclwwlkn 30108. (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 30094*	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 30095	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 30096	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 30097*	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 30098	∼ 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 30099*	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 30100*	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)))