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Theorem List for Metamath Proof Explorer - 26201-26300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisconngra1 26201* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 ConnGrph 𝐸 ↔ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘})∃𝑓𝑝 𝑓(𝑘(𝑉 PathOn 𝐸)𝑛)𝑝))

Theorem0conngra 26202 A class/graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(𝐸𝑉 → ∅ ConnGrph 𝐸)

Theorem1conngra 26203 A class/graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(𝐸𝑉 → {𝐴} ConnGrph 𝐸)

Theoremcusconngra 26204 A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(𝑉 ComplUSGrph 𝐸𝑉 ConnGrph 𝐸)

17.1.5.5  Walks as words

Syntaxcwwlk 26205 Extend class notation with Walks (of a graph) as Word over the set of vertices.
class WWalks

Syntaxcwwlkn 26206 Extend class notation with Walks (of a graph) of a fixed length as Word over the set of vertices.
class WWalksN

Definitiondf-wwlk 26207* Define the set of all 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) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlk 26036. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 26055. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
WWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒)})

Definitiondf-wwlkn 26208* Define the set of all Walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlk 26036. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))

Theoremwwlk 26209* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 WWalks 𝐸) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸)})

Theoremwwlkn 26210* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 WWalks 𝐸) ∣ (#‘𝑤) = (𝑁 + 1)})

Theoremiswwlk 26211* Properties of a word to represent a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌) → (𝑊 ∈ (𝑉 WWalks 𝐸) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸)))

Theoremiswwlkn 26212 Properties of a word to represent a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 WWalks 𝐸) ∧ (#‘𝑊) = (𝑁 + 1))))

Theoremwwlkprop 26213 Properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
(𝑃 ∈ (𝑉 WWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))

Theoremwwlknprop 26214 Properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.)
(𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑁 ∈ ℕ0𝑃 ∈ Word 𝑉)))

Theoremwwlknimp 26215* Implications for a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸))

Theoremwwlksswrd 26216 Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
(𝑉 WWalks 𝐸) ⊆ Word 𝑉

Theoremwwlkn0 26217* A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘0) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)

Theoremwlkiswwlk1 26218 The sequence of vertices in a walk is a walk as word in an undirected simple graph. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝐹(𝑉 Walks 𝐸)𝑃𝑃 ∈ (𝑉 WWalks 𝐸)))

Theoremwlkiswwlk2lem1 26219* Lemma 1 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))

Theoremwlkiswwlk2lem2 26220* Lemma 2 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       (((#‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((#‘𝑃) − 1))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))

Theoremwlkiswwlk2lem3 26221* Lemma 3 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → 𝑃:(0...(#‘𝐹))⟶𝑉)

Theoremwlkiswwlk2lem4 26222* Lemma 4 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))

Theoremwlkiswwlk2lem5 26223* Lemma 5 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸𝐹 ∈ Word dom 𝐸))

Theoremwlkiswwlk2lem6 26224* Lemma 6 for wlkiswwlk2 26225. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))

Theoremwlkiswwlk2 26225* A walk as word corresponds to the sequence of vertices in a walk in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 WWalks 𝐸) → ∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃))

Theoremwlkiswwlk 26226* A walk as word corresponds to a walk in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (∃𝑓 𝑓(𝑉 Walks 𝐸)𝑃𝑃 ∈ (𝑉 WWalks 𝐸)))

Theoremwlklniswwlkn1 26227 The sequence of vertices in a walk of length n is a walk as word of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 𝑁) → 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwlklniswwlkn2 26228* A walk of length n as word corresponds to the sequence of vertices in a walk of length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓(𝑓(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝑓) = 𝑁)))

Theoremwlklniswwlkn 26229* A walk of length n as word corresponds to a walk with length n in an undirected simple graph. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
(𝑉 USGrph 𝐸 → (∃𝑓(𝑓(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝑓) = 𝑁) ↔ 𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlkiswwlkn 26230 A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
(𝑃 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝑃 ∈ (𝑉 WWalks 𝐸))

Theoremwwlksswwlkn 26231 The walks of a fixed length as words are walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 17-Jul-2018.)
((𝑉 WWalksN 𝐸)‘𝑁) ⊆ (𝑉 WWalks 𝐸)

Theoremwwlknimpb 26232 Basic implications for a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = (𝑁 + 1)))

Theoremwwlkn0s 26233* The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
((𝑉𝑋𝐸𝑌) → ((𝑉 WWalksN 𝐸)‘0) = {𝑤 ∈ Word 𝑉 ∣ (#‘𝑤) = 1})

Theoremvfwlkniswwlkn 26234 If the edge function of a walk has length n, then the vertex function of the walk is a word representing the walk as word of length n. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑁 ∈ ℕ0 ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ ((𝑉 WWalksN 𝐸)‘𝑁))

Theorem2wlkeq 26235* Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.)
((𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))

Theoremusg2wlkeq 26236* Conditions for two walks within the same undirected simple graph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ (𝑉 Walks 𝐸) ∧ 𝐵 ∈ (𝑉 Walks 𝐸)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))

Theoremusg2wlkeq2 26237 Conditions for which two walks within the same undirected simple graph are the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
(((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) ∧ (𝑋 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑋)) = 𝑁) ∧ (𝑊 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st𝑊)) = 𝑁)) → ((2nd𝑋) = (2nd𝑊) → 𝑋 = 𝑊))

Theoremwlknwwlknfun 26238* Lemma 1 for wlknwwlknbij2 26242. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       (𝑁 ∈ ℕ0𝐹:𝑇𝑊)

Theoremwlknwwlkninj 26239* Lemma 2 for wlknwwlknbij2 26242. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlknwwlknsur 26240* Lemma 3 for wlknwwlknbij2 26242. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlknwwlknbij 26241* Lemma 4 for wlknwwlknbij2 26242. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = ((𝑉 WWalksN 𝐸)‘𝑁)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlknwwlknbij2 26242* There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}–1-1-onto→((𝑉 WWalksN 𝐸)‘𝑁))

Theoremwlknwwlknen 26243* The set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ≈ ((𝑉 WWalksN 𝐸)‘𝑁))

Theoremwlknwwlkneqs 26244* The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → (#‘{𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁}) = (#‘((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwlkiswwlkfun 26245* Lemma 1 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlkiswwlkinj 26246* Lemma 2 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlkiswwlksur 26247* Lemma 3 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 23-Jul-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlkiswwlkbij 26248* Lemma 4 for wlkiswwlkbij2 26249. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
𝑇 = {𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlkiswwlkbij2 26249* There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
((𝑉 USGrph 𝐸𝑃𝑉𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃})

Theoremwwlkeq 26250* Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.)
((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 𝑇 ∈ (𝑉 WWalks 𝐸)) → (𝑊 = 𝑇 ↔ ((#‘𝑊) = (#‘𝑇) ∧ ∀𝑖 ∈ (0..^(#‘𝑊))(𝑊𝑖) = (𝑇𝑖))))

Theoremwwlknred 26251 Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlknext 26252 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.)
((𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑆𝑉 ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)))

Theoremwwlknextbi 26253 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
(((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {( lastS ‘𝑇), 𝑆} ∈ ran 𝐸)) → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ↔ 𝑇 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)))

Theoremwwlknredwwlkn 26254* For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) → ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

Theoremwwlknredwwlkn0 26255* For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.)
((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ ran 𝐸)))

Theoremwwlkextwrd 26256* Lemma 0 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 5-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)})

Theoremwwlkextfun 26257* Lemma 1 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷𝑅)

Theoremwwlkextinj 26258* Lemma 2 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)

Theoremwwlkextsur 26259* Lemma 3 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐹:𝐷onto𝑅)

Theoremwwlkextbij0 26260* Lemma 4 for wwlkextbij 26261. (Contributed by Alexander van der Vekens, 7-Aug-2018.)
𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → 𝐹:𝐷1-1-onto𝑅)

Theoremwwlkextbij 26261* There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸})

Theoremwwlkexthasheq 26262* The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Proof shortened by AV, 4-May-2021.)
(𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ ran 𝐸)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ ran 𝐸}))

Theoremwwlkm1edg 26263 Removing the trailing edge from a walk (as word) with at least one edge results in a walk. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 2 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (𝑉 WWalks 𝐸))

Theoremdisjxwwlks 26264* Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

Theoremwwlknndef 26265 Conditions for WWalksN not being defined. (Contributed by Alexander van der Vekens, 30-Jul-2018.)
((𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = ∅)

Theoremwwlknfi 26266 The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.)
(𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)

Theoremwlknfi 26267* The number of walks of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑉 ∈ Fin) → {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∈ Fin)

Theoremwlknwwlknvbij 26268* There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑋𝑉) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})

Theoremwwlkextproplem1 26269 Lemma 1 for wwlkextprop 26272. (Contributed by Alexander van der Vekens, 31-Jul-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))       ((𝑊𝑋𝑁 ∈ ℕ0) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))

Theoremwwlkextproplem2 26270 Lemma 2 for wwlkextprop 26272. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))       ((𝑊𝑋𝑁 ∈ ℕ0) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)

Theoremwwlkextproplem3 26271* Lemma 3 for wwlkextprop 26272. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       ((𝑊𝑋 ∧ (𝑊‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)

Theoremwwlkextprop 26272* Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)})

Theoremdisjxwwlkn 26273* Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

Theoremhashwwlkext 26274* Number of walks (as words) extended by an edge as sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       (𝑉 ∈ Fin → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}) = Σ𝑦𝑌 (#‘{𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}))

17.1.5.6  Closed walks

Syntaxcclwlk 26275 Extend class notation with Closed Walks (of a graph).
class ClWalks

Syntaxcclwwlk 26276 Extend class notation with Closed Walks (of a graph) as Word over the set of vertices.
class ClWWalks

Syntaxcclwwlkn 26277 Extend class notation with Closed Walks (of a graph) of a fixed length as Word over the set of vertices.
class ClWWalksN

Definitiondf-clwlk 26278* Define the set of all Closed Walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26293! (Contributed by Alexander van der Vekens, 12-Mar-2018.)

ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Definitiondf-clwwlk 26279* 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-clwlk 26278. 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.

Notice that by this definition, a single vertex cannot be represented as closed walk, since the word <" v "> with vertex v represents the walk "vv", which is a (closed) walk of length 1 (if there is an edge/loop from v to v). Therefore, a closed walk corresponds to a closed walk as word in an undirected graph only for walks of length at least 1, see clwlkisclwwlk2 26318. (Contributed by Alexander van der Vekens, 20-Mar-2018.)

ClWWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝑒)})

Definitiondf-clwwlkn 26280* Define the set of all Closed Walks (in an undirected graph) of a fixed length n 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-clwlk 26278. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))

Theoremclwlk 26281* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWalks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Theoremisclwlk0 26282 Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremisclwlkg 26283 Generalisation of isclwlk0 26282: Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremisclwlk 26284* Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))

Theoremclwlkiswlk 26285 A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(𝐹(𝑉 ClWalks 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremclwlkswlks 26286 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝑊 ∈ (𝑉 ClWalks 𝐸) → 𝑊 ∈ (𝑉 Walks 𝐸))

Theoremclwlksarewlks 26287 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
(𝑉 ClWalks 𝐸) ⊆ (𝑉 Walks 𝐸)

Theoremwlkv0 26288 If there is a walk in an empty graph, it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(𝑊 ∈ (∅ Walks 𝐸) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Theoremwlk0 26289 There is no walk in an empty graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(∅ Walks 𝐸) = ∅

Theoremclwlk0 26290 There is no closed walk in an empty graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(∅ ClWalks 𝐸) = ∅

Theoremclwlkcomp 26291* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝑉𝑋𝐸𝑌𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (𝑉 ClWalks 𝐸) ↔ ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))

Theoremclwlkcompim 26292* Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theorem0clwlk 26293 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.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 ClWalks 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theoremclwwlk 26294* 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.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWWalks 𝐸) = {𝑤 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝐸)})

Theoremclwwlkn 26295* The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})

Theoremisclwwlk 26296* Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))

Theoremisclwwlkn 26297 Properties of a word to represent a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑊) = 𝑁)))

Theoremclwwlkprop 26298 Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))

Theoremclwwlkgt0 26299 A closed walk in an undirected graph has a length of at least 2. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))

Theoremclwwlknprop 26300 Properties of a closed walk of a fixed length as word. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
(𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))

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