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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rusgrnumwwlkb1 30001* | Induction base 1 for rusgrnumwwlk 30004. (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 30002* | 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 30003* | Induction step for rusgrnumwwlk 30004. (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 30004* |
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 30005* | 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 30004. (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 30006* | 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 30007* | 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 30008* | 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))) | ||
In general, a closed walk is an alternating sequence of vertices and edges, as defined in df-clwlks 29803: 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 14549, is also used in Definitions df-clwwlk 30010 and df-clwwlkn 30053, 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 30010, df-clwwlkn 30053 and df-clwwlknon 30116, 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 30159, 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 30070. Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 30031 or clwlkclwwlken 30040. 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 30056). 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 30116. This definition is also based on ℕ0 instead of ℕ, with (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 30120). clwwlknon1le1 30129 states that there is at most one (closed) walk of length 1 on a vertex, which would consist of a loop (see clwwlknon1loop 30126). And in a 𝐾-regular graph, there are 𝐾 closed walks of length 2 on each vertex, see clwwlknon2num 30133. | ||
Syntax | cclwwlk 30009 | Extend class notation with closed walks (in an undirected graph) as word over the set of vertices. |
class ClWWalks | ||
Definition | df-clwwlk 30010* | 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 29803. 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 30011* | 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 30012* | 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 30013 | 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 30014 | 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 30015 | 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 30016 | A closed walk of length 1 is a loop. See also clwlkl1loop 29815. (Contributed by AV, 24-Apr-2021.) |
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (♯‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺)) | ||
Theorem | clwwlkccatlem 30017* | Lemma for clwwlkccat 30018: 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 30018 | 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 30019 | 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 30020* | Lemma 1 for clwlkclwwlklem2a 30026. (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 30021* | Lemma 2 for clwlkclwwlklem2a 30026. (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 30022* | Lemma 3 for clwlkclwwlklem2a 30026. (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 30023* | Lemma 4a for clwlkclwwlklem2a 30026. (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 30024* | Lemma 4b for clwlkclwwlklem2a 30026. (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 30025* | Lemma 4 for clwlkclwwlklem2a 30026. (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 30026* | Lemma for clwlkclwwlklem2 30028. (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 30027* | Lemma 1 for clwlkclwwlk 30030. (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 30028* | Lemma 2 for clwlkclwwlk 30030. (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 30029* | Lemma 3 for clwlkclwwlk 30030. (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 30030* | 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 30031* | 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 30032* | Lemma for clwlkclwwlkf 30036. (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 30033* | Lemma 2 for clwlkclwwlkf1 30038. (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 30034* | Lemma 3 for clwlkclwwlkf1 30038. (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 30035* | Lemma for clwlkclwwlkfo 30037. (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 30036* | 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 30037* | 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 30038* | 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 30039* | 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 30040* | 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 30041* | Lemma for clwwisshclwwslem 30042. (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 30042* | Lemma for clwwisshclwws 30043. (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 30043 | 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 30044 | 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 30045 | ∼ 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 30046* | 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 30047* | 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 30048* | ∼ 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 30049* | ∼ 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 30050* | ∼ 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 30051* | ∼ 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) | ||
Syntax | cclwwlkn 30052 | 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 30053* | 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 29803. For 𝑛 = 0, the set is empty, see clwwlkn0 30056. (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 30054* | 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 30055 | 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 30056 | 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 30057 | 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 30058 | 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 30059 | 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 30060 | 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 30061 | 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 30062 | 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 30063 | 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 30064* | 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 30065* | 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 30066 | 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 30067 | 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 30068 | 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 30069 | 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 30070 | 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 30071* | 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 30072 | 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 30073 | 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 30074* | 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 30075* | Lemma 1 for clwwlkf1o 30079: 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 30076* | Lemma 2 for clwwlkf1o 30079: 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 30077* | Lemma 3 for clwwlkf1o 30079: 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 30078* | Lemma 4 for clwwlkf1o 30079: 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 30079* | 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 30080* | 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 30081* | 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 30082 | 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 30083 | 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 30084 | 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 30085 | 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 30086 | 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 30087 | 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 30088* | Lemma 1 for eleclclwwlkn 30104. (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 30089* | Lemma 2 for eleclclwwlkn 30104. (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 30090* | 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 30091 | 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 30092 | 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 30093* | 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 30094 | ∼ 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 30095* | 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 30096* | 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 30097* | ∼ 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 30098* | ∼ 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 30099* | ∼ 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 30100* | ∼ 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 |
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