P. 265 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	el2wlkonotot1 26401	A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅 ∧ 𝐶 = 𝑆 ∧ ⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶))))
Theorem	2wlkonot3v 26402	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) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
Theorem	2spthonot3v 26403	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) ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝑇 ∈ ((𝑉 × 𝑉) × 𝑉)))
Theorem	2wlkonotv 26404	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) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)))
Theorem	el2wlksoton 26405*	A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑇 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)))
Theorem	el2spthsoton 26406*	A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑇 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑇 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)))
Theorem	el2wlksot 26407*	A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑇 ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑇 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Theorem	el2pthsot 26408*	A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑇 ∈ (𝑉 2SPathsOt 𝐸) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑇 = ⟨𝑎, 𝑏, 𝑐⟩ ∧ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Theorem	el2wlksotot 26409*	A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑉 2WalksOt 𝐸) ↔ ∃𝑓∃𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
Theorem	usg2wlkonot 26410	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
Theorem	usg2wotspth 26411*	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ 𝐴 ≠ 𝐶) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓∃𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))
Theorem	2pthwlkonot 26412	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))
Theorem	2wot2wont 26413*	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.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉 2WalksOt 𝐸) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ 𝑉 (𝑥(𝑉 2WalksOnOt 𝐸)𝑦))
Theorem	2spontn0vne 26414	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) ≠ ∅ → 𝑋 ≠ 𝑌)
Theorem	usg2spthonot 26415	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸)))
Theorem	usg2spthonot0 26416	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨𝑆, 𝐵, 𝑇⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ((𝑆 = 𝐴 ∧ 𝑇 = 𝐶 ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸))))
Theorem	usg2spthonot1 26417*	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 ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸))))
Theorem	2spot2iun2spont 26418*	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) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑉 2SPathOnOt 𝐸)𝑦))
Theorem	2spotfi 26419	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 ∧ 𝐸 ∈ 𝑋) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) ∈ Fin)
17.1.6  Vertex degree
Syntax	cvdg 26420	Extend class notation with the vertex degree function.
class VDeg
Definition	df-vdgr 26421*	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 ↦ (𝑢 ∈ 𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒 ∣ 𝑢 ∈ (𝑒‘𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) = {𝑢}}))))
Theorem	vdgrfval 26422*	The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸) = (𝑢 ∈ 𝑉 ↦ ((#‘{𝑥 ∈ 𝐴 ∣ 𝑢 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑢}}))))
Theorem	vdgrval 26423*	The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) +𝑒 (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
Theorem	vdgrfival 26424*	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.)
⊢ (((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = ((#‘{𝑥 ∈ 𝐴 ∣ 𝑈 ∈ (𝐸‘𝑥)}) + (#‘{𝑥 ∈ 𝐴 ∣ (𝐸‘𝑥) = {𝑈}})))
Theorem	vdgrf 26425	The vertex degree function is a function from vertices to nonnegative integers or plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ 𝑋) → (𝑉 VDeg 𝐸):𝑉⟶(ℕ0 ∪ {+∞}))
Theorem	vdgrfif 26426	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.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝑉 VDeg 𝐸):𝑉⟶ℕ0)
Theorem	vdgr0 26427	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.)
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg ∅)‘𝑈) = 0)
Theorem	vdgrun 26428	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.)
⊢ (𝜑 → 𝐸 Fn 𝐴)    &   ⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑉 UMGrph 𝐸)    &   ⊢ (𝜑 → 𝑉 UMGrph 𝐹)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) +𝑒 ((𝑉 VDeg 𝐹)‘𝑈)))
Theorem	vdgrfiun 26429	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.)
⊢ (𝜑 → 𝐸 Fn 𝐴)    &   ⊢ (𝜑 → 𝐹 Fn 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑉 UMGrph 𝐸)    &   ⊢ (𝜑 → 𝑉 UMGrph 𝐹)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg (𝐸 ∪ 𝐹))‘𝑈) = (((𝑉 VDeg 𝐸)‘𝑈) + ((𝑉 VDeg 𝐹)‘𝑈)))
Theorem	vdgr1d 26430	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.)
⊢ (𝜑 → 𝑉 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝑈}⟩})‘𝑈) = 2)
Theorem	vdgr1b 26431	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.)
⊢ (𝜑 → 𝑉 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝑈, 𝐵}⟩})‘𝑈) = 1)
Theorem	vdgr1c 26432	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.)
⊢ (𝜑 → 𝑉 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝐵, 𝑈}⟩})‘𝑈) = 1)
Theorem	vdgr1a 26433	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.)
⊢ (𝜑 → 𝑉 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ 𝑈)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ≠ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑉 VDeg {⟨𝐴, {𝐵, 𝐶}⟩})‘𝑈) = 0)
Theorem	vdusgraval 26434*	The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑈) = (#‘{𝑥 ∈ dom 𝐸 ∣ 𝑈 ∈ (𝐸‘𝑥)}))
Theorem	vdusgra0nedg 26435*	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 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 ∈ Fin) → (((𝑉 VDeg 𝐸)‘𝑈) = 0 → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ ran 𝐸))
Theorem	vdgrnn0pnf 26436	The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑋 ∈ 𝑉) → ((𝑉 VDeg 𝐸)‘𝑋) ∈ (ℕ0 ∪ {+∞}))
Theorem	usgfidegfi 26437*	In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin) → ∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ∈ ℕ0)
Theorem	usgfiregdegfi 26438*	In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0))
Theorem	hashnbgravd 26439	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 𝐸 ∧ 𝑈 ∈ 𝑉 ∧ 𝐸 ∈ Fin) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))
Theorem	hashnbgravdg 26440	The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 26439. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑈)) = ((𝑉 VDeg 𝐸)‘𝑈))
Theorem	nbhashnn0 26441	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 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) ∈ ℕ0)
Theorem	nbhashuvtx1 26442	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 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → ((𝑀 ∈ 𝑉 ∧ 𝑀 ≠ 𝑁) → 𝑀 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁))))
Theorem	nbhashuvtx 26443	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 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → ((#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1) → 𝑁 ∈ (𝑉 UnivVertex 𝐸)))
Theorem	uvtxhashnb 26444	A universal vertex has 𝑛 − 1 neighbors in a graph with 𝑛 vertices, a biconditional version of uvtxnm1nbgra 26022. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)))
Theorem	usgravd0nedg 26445*	If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge, analogous to vdusgra0nedg 26435. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑈 ∈ 𝑉) → (((𝑉 VDeg 𝐸)‘𝑈) = 0 → ¬ ∃𝑣 ∈ 𝑉 {𝑈, 𝑣} ∈ ran 𝐸))
Theorem	usgravd00 26446*	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 → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 0 → 𝐸 = ∅))
Theorem	usgrauvtxvdbi 26447	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 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝐾 ∈ 𝑉) → (𝐾 ∈ (𝑉 UnivVertex 𝐸) ↔ ((𝑉 VDeg 𝐸)‘𝐾) = ((#‘𝑉) − 1)))
Theorem	vdiscusgra 26448*	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 𝐸 ∧ 𝑉 ∈ Fin) → (∀𝑣 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = ((#‘𝑉) − 1) → 𝑉 ComplUSGrph 𝐸))
17.1.7  Regular graphs
17.1.7.1  Definition and basic properties
Syntax	crgra 26449	Extend class notation to include the class of all regular graphs.
class RegGrph
Syntax	crusgra 26450	Extend class notation to include the class of all regular undirected simple graphs.
class RegUSGrph
Definition	df-rgra 26451*	Define the class of k-regular "graphs". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
⊢ RegGrph = {⟨⟨𝑣, 𝑒⟩, 𝑘⟩ ∣ (𝑘 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑣 ((𝑣 VDeg 𝑒)‘𝑝) = 𝑘)}
Definition	df-rusgra 26452*	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)}
Theorem	isrgra 26453*	The property of being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
Theorem	isrusgra 26454*	The property of being a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾)))
Theorem	rgraprop 26455*	The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegGrph 𝐾 → (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
Theorem	rusgraprop 26456*	The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 ((𝑉 VDeg 𝐸)‘𝑝) = 𝐾))
Theorem	rusgrargra 26457	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)
Theorem	rusisusgra 26458	Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → 𝑉 USGrph 𝐸)
Theorem	isrusgusrg 26459	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.)
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐾 ∈ 𝑍) → (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ↔ (𝑉 USGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegGrph 𝐾)))
Theorem	isrusgusrgcl 26460	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) ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐺 RegGrph 𝐾)))
Theorem	isrgrac 26461*	The property of being a k-regular graph represented by a class. (Contributed by AV, 3-Jan-2020.)
⊢ ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGrph 𝐾 ↔ (𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
Theorem	isrusgrac 26462*	The property of being a k-regular undirected simple graph represented by a class. (Contributed by AV, 3-Jan-2020.)
⊢ ((𝐺 ∈ (𝑋 × 𝑌) ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGrph 𝐾 ↔ (𝐺 ∈ USGrph ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ (1st ‘𝐺)(( VDeg ‘𝐺)‘𝑝) = 𝐾)))
Theorem	0egra0rgra 26463	A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
⊢ ∀𝑣⟨𝑣, ∅⟩ RegGrph 0
Theorem	0vgrargra 26464*	A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
⊢ (𝐸 ∈ 𝑉 → ∀𝑘 ∈ ℕ0 ⟨∅, 𝐸⟩ RegGrph 𝑘)
Theorem	cusgraisrusgra 26465	A complete undirected simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
⊢ ((𝑉 ComplUSGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1))
Theorem	0eusgraiff0rgra 26466	An undirected simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
⊢ (𝑉 USGrph 𝐸 → (⟨𝑉, 𝐸⟩ RegGrph 0 ↔ 𝐸 = ∅))
Theorem	cusgraiffrusgra 26467	A finite undirected simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGrph allowed for 𝑘 ∈ ℤ, then the assumption 𝑉 ≠ ∅ could be removed. Furthermore, if the definition of RegGrph also allowed for 𝑘 ∈ (ℤ ∪ {+∞}), then the theorem would also hold for inifinite graphs. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
⊢ ((𝑉 USGrph 𝐸 ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (𝑉 ComplUSGrph 𝐸 ↔ ⟨𝑉, 𝐸⟩ RegUSGrph ((#‘𝑉) − 1)))
Theorem	0eusgraiff0rgracl 26468	An undirected simple graph represented by a class is 0-regular iff it has no edges. (Contributed by AV, 3-Jan-2020.)
⊢ (𝐺 ∈ USGrph → (𝐺 RegGrph 0 ↔ (Edges‘𝐺) = ∅))
Theorem	rusgraprop2 26469*	The properties of a k-regular undirected simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑝)) = 𝐾))
Theorem	rusgraprop3 26470*	The properties of a k-regular undirected simple graph expressed with edges. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ 𝑉 (#‘{𝑛 ∈ 𝑉 ∣ {𝑝, 𝑛} ∈ ran 𝐸}) = 𝐾))
Theorem	rusgraprop4 26471*	The properties of a k-regular undirected simple graph expressed with trailing edges of walks (as words). (Contributed by Alexander van der Vekens, 2-Aug-2018.)
⊢ (⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 → (𝑉 USGrph 𝐸 ∧ 𝐾 ∈ ℕ0 ∧ ∀𝑝 ∈ Word 𝑉(𝑝 ≠ ∅ → (#‘{𝑛 ∈ 𝑉 ∣ {( lastS ‘𝑝), 𝑛} ∈ ran 𝐸}) = 𝐾)))
Theorem	rusgrasn 26472	If a k-regular undirected simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
⊢ (((#‘𝑉) = 1 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → 𝐾 = 0)
17.1.7.2  Walks in regular graphs
Theorem	rusgranumwwlkl1 26473*	In a k-regular graph, the number of walks of length 1 represented as words (thus the number of edges) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Proof shortened by AV, 4-May-2021.)
⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸)}) = 𝐾)
Theorem	rusgranumwlkl1 26474*	In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘1) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Theorem	rusgranumwlklem0 26475*	Lemma 0 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)})
Theorem	rusgranumwlklem1 26476*	Lemma 1 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    ⇒   ⊢ (𝑅 ∈ (𝑊‘𝑁) → (𝑅 ∈ (𝑉 Walks 𝐸) ∧ (#‘(1st ‘𝑅)) = 𝑁))
Theorem	rusgranumwlklem2 26477*	Lemma 2 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑊‘𝑁) ∣ ((2nd ‘𝑤)‘0) = 𝑃}))
Theorem	rusgranumwlklem3 26478*	Lemma 3 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}))
Theorem	rusgranumwlklem4 26479*	Lemma 4 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Proof shortened by AV, 5-May-2021.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}))
Theorem	rusgranumwlkb0 26480*	Induction base 0 for rusgranumwlk 26484. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
Theorem	rusgranumwlkb1 26481*	Induction base 1 for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿1) = 𝐾)
Theorem	rusgra0edg 26482*	Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
Theorem	rusgranumwlks 26483*	Induction step for rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Theorem	rusgranumwlk 26484*	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. We denote with (𝑊‘𝑛) the set of walks with length n (in a given undirected simple graph) and with (𝑣𝐿𝑛) the number of walks with length n starting at the vertex v. 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.". Because of the k-regularity, the walk can be continued in k different ways at each vertex in the walk, therefore n times. This theorem even holds for n=0: then the walk consists only of 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.)
⊢ 𝑊 = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st ‘𝑐)) = 𝑛})    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑊‘𝑛) ∣ ((2nd ‘𝑤)‘0) = 𝑣}))    ⇒   ⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))
Theorem	rusgranumwlkg 26485*	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 only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. Closed form of rusgranumwlk 26484. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁))
Theorem	rusgranumwwlkg 26486*	In a k-regular graph, the number of walks (represented by words) of a fixed length n from a fixed vertex is k to the power of n. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
⊢ ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁))
Theorem	clwlknclwlkdifs 26487	The set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between the set of walks of length n starting with this vertex and the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
⊢ 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}    ⇒   ⊢ 𝐴 = ({𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
Theorem	clwlknclwlkdifnum 26488*	In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
⊢ 𝐴 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}    ⇒   ⊢ (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))
17.2  Eulerian paths and the Konigsberg Bridge problem
17.2.1  Eulerian paths
Syntax	ceup 26489	Extend class notation with Eulerian paths.
class EulPaths
Definition	df-eupa 26490*	Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom 𝑒 ∧ 𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓‘𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝‘𝑘)}))})
Theorem	releupa 26491	The set (𝑉 EulPaths 𝐸) of all Eulerian paths on ⟨𝑉, 𝐸⟩ is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ Rel (𝑉 EulPaths 𝐸)
Theorem	iseupa 26492*	The property "⟨𝐹, 𝑃⟩ is an Eulerian path on the graph ⟨𝑉, 𝐸⟩". An Eulerian path is defined as bijection 𝐹 from the edges to a set 1...𝑁 a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 1 ≤ 𝑘 ≤ 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘 − 1) to 𝑃(𝑘). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (dom 𝐸 = 𝐴 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}))))
Theorem	eupagra 26493	If an eulerian path exists, then ⟨𝑉, 𝐸⟩ is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝑉 UMGrph 𝐸)
Theorem	eupai 26494*	Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝐸 Fn 𝐴) → (((#‘𝐹) ∈ ℕ0 ∧ 𝐹:(1...(#‘𝐹))–1-1-onto→𝐴 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃‘𝑘)}))
Theorem	eupatrl 26495*	An Eulerian path is a trail. Unfortunately, the edge function 𝐹 of an Eulerian path has the domain (1...(#‘𝐹)), whereas the edge functions of all kinds of walks defined here have the domain (0..^(#‘𝐹)) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 26162, fargshiftfv 26163, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
⊢ 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))    ⇒   ⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝐺(𝑉 Trails 𝐸)𝑃)
Theorem	eupacl 26496	An Eulerian path has length #(𝐹), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → (#‘𝐹) ∈ ℕ0)
Theorem	eupaf1o 26497	The 𝐹 function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝐸 Fn 𝐴) → 𝐹:(1...(#‘𝐹))–1-1-onto→𝐴)
Theorem	eupafi 26498	Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝐸 Fn 𝐴) → 𝐴 ∈ Fin)
Theorem	eupapf 26499	The 𝑃 function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (𝐹(𝑉 EulPaths 𝐸)𝑃 → 𝑃:(0...(#‘𝐹))⟶𝑉)
Theorem	eupaseg 26500	The 𝑁-th edge in an eulerian path is the edge from 𝑃(𝑁 − 1) to 𝑃(𝑁). (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐹(𝑉 EulPaths 𝐸)𝑃 ∧ 𝑁 ∈ (1...(#‘𝐹))) → (𝐸‘(𝐹‘𝑁)) = {(𝑃‘(𝑁 − 1)), (𝑃‘𝑁)})