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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremhashclwwlkn 26101* The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (#‘𝑊) = ((#‘(𝑊 / )) · 𝑁))

Theoremclwwlkndivn 26102 The size of the set of closed walks (defined as words) of length n is divisible by n. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘((𝑉 ClWWalksN 𝐸)‘𝑁)))

Theoremwlklenvp1 26103 The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (#‘𝑃) = ((#‘𝐹) + 1))

Theoremwlklenvclwlk 26104 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝐹) = (#‘𝑊)))

Theoremclwlkfclwwlk2wrd 26105 The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶𝐵 ∈ Word 𝑉)

Theoremclwlkfclwwlk1hashn 26106* The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)

Theoremclwlkfclwwlk1hash 26107* The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))

Theoremclwlkfclwwlk2sswd 26108* The size of a subword of the second component of a closed walk with length of the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))

Theoremclwlkfclwwlk 26109* There is a function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkfoclwwlk 26110* There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 30-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkf1clwwlklem1 26111* Lemma 1 for clwlkf1clwwlklem 26114. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))

Theoremclwlkf1clwwlklem2 26112* Lemma 2 for clwlkf1clwwlklem 26114. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))

Theoremclwlkf1clwwlklem3 26113* Lemma 3 for clwlkf1clwwlklem 26114. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (2nd𝑊) ∈ Word 𝑉)

Theoremclwlkf1clwwlklem 26114* Lemma for clwlkf1clwwlk 26115. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))

Theoremclwlkf1clwwlk 26115* There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkf1oclwwlk 26116* There is a one-to-one onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1-onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlksizeeq 26117* The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Proof shortened by AV, 4-May-2021.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (#‘((𝑉 ClWWalksN 𝐸)‘𝑁)) = (#‘{𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))

Theoremclwlkndivn 26118* The size of the set of closed walks of length n is divisible by n. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘{𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))

16.1.5.7  Walks/paths of length 2 as ordered triples

Syntaxc2wlkot 26119 Extend class notation with walks (of a graph) of length 2 as ordered triple.
class 2WalksOt

Syntaxc2wlkonot 26120 Extend class notation with walks between two vertices (within a graph) of length 2 as ordered triple.
class 2WalksOnOt

Syntaxc2spthot 26121 Extend class notation with paths (of a graph) of length 2 as ordered triple.
class 2SPathsOt

Syntaxc2pthonot 26122 Extend class notation with simple paths between two vertices (within a graph) of length 2 as ordered triple.
class 2SPathOnOt

Definitiondf-2wlkonot 26123* Define the collection of walks of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
2WalksOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Definitiondf-2wlksot 26124* Define the collection of all walks of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
2WalksOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)})

Definitiondf-2spthonot 26125* Define the collection of simple paths of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
2SPathOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 SPathOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Definitiondf-2spthsot 26126* Define the collection of all simple paths of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
2SPathsOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)})

Theoremel2wlkonotlem 26127 Lemma for el2wlkonot 26134. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)

Theoremis2wlkonot 26128* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Theoremis2spthonot 26129* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2SPathOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 SPathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Theorem2wlkonot 26130* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})

Theorem2spthonot 26131* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})

Theorem2wlksot 26132* The set of walks of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})

Theorem2spthsot 26133* The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2SPathsOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})

Theoremel2wlkonot 26134* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2spthonot 26135* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2spthonot0 26136* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))

Theoremel2wlkonotot0 26137* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2wlkonotot 26138* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

Theoremel2wlkonotot1 26139 A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))

Theorem2wlkonot3v 26140 If an ordered triple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
(𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉) ∧ 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))

Theorem2spthonot3v 26141 If an ordered triple represents a simple path of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐶𝑉) ∧ 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))

Theorem2wlkonotv 26142 If an ordered tripple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
(⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)))

Theoremel2wlksoton 26143* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))

Theoremel2spthsoton 26144* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉 𝑇 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)))

Theoremel2wlksot 26145* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑇 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Theoremel2pthsot 26146* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑇 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑇 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))

Theoremel2wlksotot 26147* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

Theoremusg2wlkonot 26148 A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to prove this the cases 𝐴 = 𝐵 and/or 𝐵 = 𝐶 must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Theoremusg2wotspth 26149* A walk of length 2 between two different vertices as ordered triple corresponds to a simple path of length 2 in an undirected simple graph. (Contributed by Alexander van der Vekens, 16-Feb-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ 𝐴𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

Theorem2pthwlkonot 26150 For two different vertices, a walk of length 2 between these vertices as ordered triple is a simple path of length 2 between these vertices as ordered triple in an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))

Theorem2wot2wont 26151* The set of (simple) paths of length 2 (in a graph) is the set of (simple) paths of length 2 between any two different vertices. (Contributed by Alexander van der Vekens, 27-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = 𝑥𝑉 𝑦𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))

Theorem2spontn0vne 26152 If the set of simple paths of length 2 between two vertices (in a graph) is not empty, the two vertices must be not equal. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
((𝑋(𝑉 2SPathOnOt 𝐸)𝑌) ≠ ∅ → 𝑋𝑌)

Theoremusg2spthonot 26153 A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝐴𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))

Theoremusg2spthonot0 26154 A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴𝑇 = 𝐶𝐴𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))

Theoremusg2spthonot1 26155* A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))

Theorem2spot2iun2spont 26156* The set of simple paths of length 2 (in a graph) is the double union of the simple paths of length 2 between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 2SPathsOt 𝐸) = 𝑥𝑉 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑉 2SPathOnOt 𝐸)𝑦))

Theorem2spotfi 26157 In a finite graph, the set of simple paths of length 2 between two vertices (as ordered triples) is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
(((𝑉 ∈ Fin ∧ 𝐸𝑋) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)

16.1.6  Vertex degree

Syntaxcvdg 26158 Extend class notation with the vertex degree function.
class VDeg

Definitiondf-vdgr 26159* Define the vertex degree function (for an undirected multigraph). To be appropriate for multigraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition +𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))

Theoremvdgrfval 26160* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑢}}))))

Theoremvdgrval 26161* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
(((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))

Theoremvdgrfival 26162* The value of the vertex degree function (for graphs of finite size). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
(((𝑉𝑊𝐸 Fn 𝐴𝐴 ∈ Fin) ∧ 𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐸𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐸𝑥) = {𝑈}})))

Theoremvdgrf 26163 The vertex degree function is a function from vertices to nonnegative integers or plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉𝑊𝐸 Fn 𝐴𝐴𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))

Theoremvdgrfif 26164 The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
((𝑉𝑊𝐸 Fn 𝐴𝐴 ∈ Fin) → (𝑉 VDeg 𝐸):𝑉⟶ℕ0)

Theoremvdgr0 26165 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
((𝑉𝑊𝑈𝑉) → ((𝑉 VDeg ∅)‘𝑈) = 0)

Theoremvdgrun 26166 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑌)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UMGrph 𝐸)    &   (𝜑𝑉 UMGrph 𝐹)    &   (𝜑𝑈𝑉)       (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) +𝑒 ((𝑉 VDeg 𝐹)‘𝑈)))

Theoremvdgrfiun 26167 The degree of a vertex in the union of two graphs (of finite size) on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UMGrph 𝐸)    &   (𝜑𝑉 UMGrph 𝐹)    &   (𝜑𝑈𝑉)       (𝜑 → ((𝑉 VDeg (𝐸𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)))

Theoremvdgr1d 26168 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
(𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑈𝑉)       (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝑈}⟩})‘𝑈) = 2)

Theoremvdgr1b 26169 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
(𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵𝑈)       (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝑈, 𝐵}⟩})‘𝑈) = 1)

Theoremvdgr1c 26170 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
(𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵𝑈)       (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝐵, 𝑈}⟩})‘𝑈) = 1)

Theoremvdgr1a 26171 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
(𝜑𝑉 ∈ V)    &   (𝜑𝐴 ∈ V)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵𝑈)    &   (𝜑𝐶𝑉)    &   (𝜑𝐶𝑈)       (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝐵, 𝐶}⟩})‘𝑈) = 0)

Theoremvdusgraval 26172* The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸𝑈 ∈ (𝐸𝑥)}))

Theoremvdusgra0nedg 26173* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉𝐸 ∈ Fin) → (((𝑉 VDeg 𝐸)‘𝑈) = 0 → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ ran 𝐸))

Theoremvdgrnn0pnf 26174 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
((𝑉 USGrph 𝐸𝑋𝑉) → ((𝑉 VDeg 𝐸)‘𝑋) ∈ (ℕ0 ∪ {+∞}))

Theoremusgfidegfi 26175* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0)

Theoremusgfiregdegfi 26176* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾𝐾 ∈ ℕ0))

Theoremhashnbgravd 26177 The size of the set of the neighbors of a vertex is the vertex degree of this vertex. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉𝐸 ∈ Fin) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))

Theoremhashnbgravdg 26178 The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 26177. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))

Theoremnbhashnn0 26179 The number of the neighbors of a vertex in a finite undirected simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℕ0)

Theoremnbhashuvtx1 26180 If the number of the neighbors of a vertex in a finite graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))

Theoremnbhashuvtx 26181 If the number of the neighbors of a vertex in a graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑁 ∈ (𝑉 UnivVertex 𝐸)))

Theoremuvtxhashnb 26182 A universal vertex has 𝑛 − 1 neighbors in a graph with 𝑛 vertices, a biconditional version of uvtxnm1nbgra 25760. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)))

Theoremusgravd0nedg 26183* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge, analogous to vdusgra0nedg 26173. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
((𝑉 USGrph 𝐸𝑈𝑉) → (((𝑉 VDeg 𝐸)‘𝑈) = 0 → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ ran 𝐸))

Theoremusgravd00 26184* If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
(𝑉 USGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅))

Theoremusgrauvtxvdbi 26185 In a finite undirected simple graph with n vertices a vertex is universal if the vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝐾𝑉) → (𝐾 ∈ (𝑉 UnivVertex 𝐸) ↔ ((𝑉 VDeg 𝐸)‘𝐾) = ((#‘𝑉) − 1)))

Theoremvdiscusgra 26186* In a finite complete undirected simple graph with n vertices every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1) → 𝑉 ComplUSGrph 𝐸))

16.1.7  Regular graphs

16.1.7.1  Definition and basic properties

Syntaxcrgra 26187 Extend class notation to include the class of all regular graphs.
class RegGrph

Syntaxcrusgra 26188 Extend class notation to include the class of all regular undirected simple graphs.
class RegUSGrph

Definitiondf-rgra 26189* Define the class of k-regular "graphs". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
RegGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}

Definitiondf-rusgra 26190* Define the class of k-regular undirected simple graphs. (Contributed by Alexander van der Vekens, 6-Jul-2018.)
RegUSGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑣 USGrph 𝑒 ∧ ⟨𝑣, 𝑒⟩ RegGrph 𝑘)}

Theoremisrgra 26191* The property of being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))

Theoremisrusgra 26192* The property of being a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))

Theoremrgraprop 26193* The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))

Theoremrusgraprop 26194* The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸𝐾 ∈ ℕ0 ∧ ∀𝑝𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))

Theoremrusgrargra 26195 A k-regular undirected simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → ⟨𝑉, 𝐸⟩ RegGrph 𝐾)

Theoremrusisusgra 26196 Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
(⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 USGrph 𝐸)

Theoremisrusgusrg 26197 A graph is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 3-Jan-2020.)
((𝑉𝑋𝐸𝑌𝐾𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))

Theoremisrusgusrgcl 26198 A graph represented by a class is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 2-Jan-2020.)
((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))

Theoremisrgrac 26199* The property of being a k-regular graph represented by a class. (Contributed by AV, 3-Jan-2020.)
((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))

Theoremisrusgrac 26200* The property of being a k-regular undirected simple graph represented by a class. (Contributed by AV, 3-Jan-2020.)
((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))

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