P. 301 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vtxdginducedm1lem2 30001*	Lemma 2 for vtxdginducedm1 30004: the domain of the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ dom (iEdg‘𝑆) = 𝐼
Theorem	vtxdginducedm1lem3 30002*	Lemma 3 for vtxdginducedm1 30004: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ (𝐻 ∈ 𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸‘𝐻))
Theorem	vtxdginducedm1lem4 30003*	Lemma 4 for vtxdginducedm1 30004. (Contributed by AV, 17-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (𝑊 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘 ∈ 𝐽 ∣ (𝐸‘𝑘) = {𝑊}}) = 0)
Theorem	vtxdginducedm1 30004*	The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 17-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)}))
Theorem	vtxdginducedm1fi 30005*	The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 of finite size obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 18-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑙)})))
Theorem	finsumvtxdg2ssteplem1 30006*	Lemma for finsumvtxdg2sstep 30010. (Contributed by AV, 15-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘𝐽)))
Theorem	finsumvtxdg2ssteplem2 30007*	Lemma for finsumvtxdg2sstep 30010. (Contributed by AV, 12-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})))
Theorem	finsumvtxdg2ssteplem3 30008*	Lemma for finsumvtxdg2sstep 30010. (Contributed by AV, 19-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖 ∈ 𝐽 ∣ 𝑣 ∈ (𝐸‘𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}})) = (♯‘𝐽))
Theorem	finsumvtxdg2ssteplem4 30009*	Lemma for finsumvtxdg2sstep 30010. (Contributed by AV, 12-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    &   ⊢ 𝐽 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∈ (𝐸‘𝑖)}    ⇒   ⊢ ((((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸‘𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))
Theorem	finsumvtxdg2sstep 30010*	Induction step of finsumvtxdg2size 30011: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    &   ⊢ 𝐾 = (𝑉 ∖ {𝑁})    &   ⊢ 𝐼 = {𝑖 ∈ dom 𝐸 ∣ 𝑁 ∉ (𝐸‘𝑖)}    &   ⊢ 𝑃 = (𝐸 ↾ 𝐼)    &   ⊢ 𝑆 = ⟨𝐾, 𝑃⟩    ⇒   ⊢ (((𝐺 ∈ UPGraph ∧ 𝑁 ∈ 𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣 ∈ 𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
Theorem	finsumvtxdg2size 30011*	The sum of the degrees of all vertices of a finite pseudograph of finite size is twice the size of the pseudograph. See equation (1) in section I.1 in [Bollobas] p. 4. Here, the "proof" is simply the statement "Since each edge has two endvertices, the sum of the degrees is exactly twice the number of edges". The formal proof of this theorem (for pseudographs) is much more complicated, taking also the used auxiliary theorems into account. The proof for a (finite) simple graph (see fusgr1th 30012) would be shorter, but nevertheless still laborious. Although this theorem would hold also for infinite pseudographs and pseudographs of infinite size, the proof of this most general version (see theorem "sumvtxdg2size" below) would require many more auxiliary theorems (e.g., the extension of the sum Σ over an arbitrary set). I dedicate this theorem and its proof to Norman Megill, who deceased too early on December 9, 2021. This proof is an example for the rigor which was the main motivation for Norman Megill to invent and develop Metamath, see section 1.1.6 "Rigor" on page 19 of the Metamath book: "... it is usually assumed in mathematical literature that the person reading the proof is a mathematician familiar with the specialty being described, and that the missing steps are obvious to such a reader or at least the reader is capable of filling them in." I filled in the missing steps of Bollobas' proof as Norm would have liked it... (Contributed by Alexander van der Vekens, 19-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))
Theorem	fusgr1th 30012*	The sum of the degrees of all vertices of a finite simple graph is twice the size of the graph. See equation (1) in section I.1 in [Bollobas] p. 4. Also known as the "First Theorem of Graph Theory" (see https://charlesreid1.com/wiki/First_Theorem_of_Graph_Theory). (Contributed by AV, 26-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → Σ𝑣 ∈ 𝑉 (𝐷‘𝑣) = (2 · (♯‘𝐼)))
Theorem	finsumvtxdgeven 30013*	The sum of the degrees of all vertices of a finite pseudograph of finite size is even. See equation (2) in section I.1 in [Bollobas] p. 4, where it is also called the handshaking lemma. (Contributed by AV, 22-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ Σ𝑣 ∈ 𝑉 (𝐷‘𝑣))
Theorem	vtxdgoddnumeven 30014*	The number of vertices of odd degree is even in a finite pseudograph of finite size. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 22-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑉 ∈ Fin ∧ 𝐼 ∈ Fin) → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}))
Theorem	fusgrvtxdgonume 30015*	The number of vertices of odd degree is even in a finite simple graph. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 27-Dec-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝐺 ∈ FinUSGraph → 2 ∥ (♯‘{𝑣 ∈ 𝑉 ∣ ¬ 2 ∥ (𝐷‘𝑣)}))
17.2.11  Regular graphs With df-rgr 30018 and df-rusgr 30019, k-regularity of a (simple) graph is defined as predicate RegGraph resp. RegUSGraph. Instead of defining a predicate, an alternative could have been to define a function that maps an extended nonnegative integer to the class of "graphs" in which every vertex has the extended nonnegative integer as degree: RegGraph = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘}). This function, however, would not be defined at least for 𝑘 = 0 (see rgrx0nd 30055), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 30053). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).
Syntax	crgr 30016	Extend class notation to include the class of all regular graphs.
class RegGraph
Syntax	crusgr 30017	Extend class notation to include the class of all regular simple graphs.
class RegUSGraph
Definition	df-rgr 30018*	Define the "k-regular" predicate, which is true for a "graph" being k-regular: read 𝐺 RegGraph 𝐾 as "𝐺 is 𝐾-regular" or "𝐺 is a 𝐾-regular graph". Note that 𝐾 is allowed to be positive infinity (𝐾 ∈ ℕ0*), as proposed by GL. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}
Definition	df-rusgr 30019*	Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read 𝐺 RegUSGraph 𝐾 as 𝐺 is a 𝐾-regular simple graph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}
Theorem	isrgr 30020*	The property of a class being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
Theorem	rgrprop 30021*	The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
Theorem	isrusgr 30022	The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
Theorem	rusgrprop 30023	The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))
Theorem	rusgrrgr 30024	A k-regular simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 RegGraph 𝐾)
Theorem	rusgrusgr 30025	A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ (𝐺 RegUSGraph 𝐾 → 𝐺 ∈ USGraph)
Theorem	finrusgrfusgr 30026	A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
Theorem	isrusgr0 30027*	The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐾 ∈ 𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
Theorem	rusgrprop0 30028*	The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
Theorem	usgreqdrusgr 30029*	If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)
Theorem	fusgrregdegfi 30030*	In a nonempty finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 19-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐾 ∈ ℕ0))
Theorem	fusgrn0eqdrusgr 30031*	If all vertices in a nonempty finite simple graph have the same (finite) degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐷 = (VtxDeg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 → 𝐺 RegUSGraph 𝐾))
Theorem	frusgrnn0 30032	In a nonempty finite k-regular simple graph, the degree of each vertex is finite. (Contributed by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
Theorem	0edg0rgr 30033	A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0)
Theorem	uhgr0edg0rgr 30034	A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
⊢ ((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0)
Theorem	uhgr0edg0rgrb 30035	A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
⊢ (𝐺 ∈ UHGraph → (𝐺 RegGraph 0 ↔ (Edg‘𝐺) = ∅))
Theorem	usgr0edg0rusgr 30036	A simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 19-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
⊢ (𝐺 ∈ USGraph → (𝐺 RegUSGraph 0 ↔ (Edg‘𝐺) = ∅))
Theorem	0vtxrgr 30037*	A null graph (with no vertices) is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘)
Theorem	0vtxrusgr 30038*	A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ ((𝐺 ∈ 𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)
Theorem	0uhgrrusgr 30039*	The null graph as hypergraph is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ ((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)
Theorem	0grrusgr 30040	The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.)
⊢ ∀𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘
Theorem	0grrgr 30041	The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
⊢ ∀𝑘 ∈ ℕ0* ∅ RegGraph 𝑘
Theorem	cusgrrusgr 30042	A complete simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((♯‘𝑉) − 1))
Theorem	cusgrm1rusgr 30043	A finite simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGraph was allowed for 𝑘 ∈ ℤ, then the assumption 𝑉 ≠ ∅ could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (𝐺 ∈ ComplUSGraph ↔ 𝐺 RegUSGraph ((♯‘𝑉) − 1)))
Theorem	rusgrpropnb 30044*	The properties of a k-regular simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 26-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
Theorem	rusgrpropedg 30045*	The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
Theorem	rusgrpropadjvtx 30046*	The properties of a k-regular simple graph expressed with adjacent vertices. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 27-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
Theorem	rusgrnumwrdl2 30047*	In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
Theorem	rusgr1vtxlem 30048*	Lemma for rusgr1vtx 30049. (Contributed by AV, 27-Dec-2020.)
⊢ (((∀𝑣 ∈ 𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) ∧ (𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0)
Theorem	rusgr1vtx 30049	If a k-regular simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.)
⊢ (((♯‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)
Theorem	rgrusgrprc 30050*	The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V
Theorem	rusgrprc 30051	The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∣ 𝑔 RegUSGraph 0} ∉ V
Theorem	rgrprc 30052	The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∣ 𝑔 RegGraph 0} ∉ V
Theorem	rgrprcx 30053*	The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V
Theorem	rgrx0ndm 30054*	0 is not in the domain of the potentially alternative definition of the sets of k-regular graphs for each extended nonnegative integer k. (Contributed by AV, 28-Dec-2020.)
⊢ 𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘})    ⇒   ⊢ 0 ∉ dom 𝑅
Theorem	rgrx0nd 30055*	The potentially alternatively defined k-regular graphs is not defined for k=0. (Contributed by AV, 28-Dec-2020.)
⊢ 𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘})    ⇒   ⊢ (𝑅‘0) = ∅
17.3  Walks, paths and cycles A "walk" in a graph is usually defined for simple graphs, multigraphs or even pseudographs as "alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see definition of [Bollobas] p. 4, or "A walk (of length k) in a graph is a nonempty alternating sequence v0 e0 v1 e1 ... e(k-1) vk of vertices and edges in G such that ei = { vi , vi+1 } for all i < k.", see definition of [Diestel] p. 10. Formalizing these definitions (mainly by representing the indexed vertices and edges by functions), a walk is represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges (e is a third function enumerating the edges within the graph, not within the walk), and p enumerates the vertices, see df-wlks 30060. Hence a walk (of length n) is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Alternatively, one could define a walk as a function 𝑤:(0...(2 · 𝑛))⟶((Edg‘𝐺) ∪ (Vtx‘𝐺)) such that for all 0 ≤ 𝑘 ≤ 𝑛, (𝑤‘(2 · 𝑘)) ∈ (Vtx‘𝐺) and for all 0 ≤ 𝑘 ≤ (𝑛 − 1), (𝑤‘((2 · 𝑘) + 1)) ∈ (Edg‘𝐺) and {(𝑤‘(2 · 𝑘)), (𝑤‘((2 · 𝑘) + 2))} ⊆ (𝑤‘((2 · 𝑘) + 1)). Based on our definition of Walks, the class of all walks, more restrictive constructs are defined: * Trails (df-trls 30155): A "walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5, i.e., f(i) =/= f(j) if i =/= j. * Paths (df-pths 30179): A path is a walk whose vertices except the first and the last vertex are distinct, i.e., p(i) =/= p(j) if i < j, except possibly when i = 0 and j = n. * SPaths (simple paths, df-spths 30180): A simple path "is a walk with distinct vertices.", see Notation of [Bollobas] p. 5, i.e., p(i) =/= p(j) if i =/= j. * ClWalks (closed walks, df-clwlks 30238): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n). * Circuits (df-crcts 30253): "A trail whose endvertices coincide (a closed trail) is called a circuit." (see Definition of [Bollobas] p. 5), i.e., f(i) =/= f(j) if i =/= j and p(0) = p(n). Equivalently, a circuit is a closed walk with distinct edges. * Cycles (df-cycls 30254): A path whose endvertices coincide (a closed path) is called a cycle, i.e., p(i) =/= p(j) if i =/= j, except i = 0 and j = n, and p(0) = p(n). Equivalently, a cycle is a closed walk with distinct vertices. * EulerPaths (Eulerian paths, df-eupth 30679): An Eulerian path is "a trail containing all edges [of the graph]" (see definition in [Bollobas] p. 16), i.e., f(i) =/= f(j) if i =/= j and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Note, however, that an Eulerian path needs not be a path. * Eulerian circuit: An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), i.e., f(i) =/= f(j) if i =/= j, p(0) = p(n) and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Hierarchy of all kinds of walks (apply ssriv 3938 and elopabran 5544 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 30241, see clwlkwlk 30242 and clwlkswks 30243): * Trails are walks (trliswlk 30160): (Trails‘𝐺) ⊆ (Walks‘𝐺) * Paths are trails (pthistrl 30188): (Paths‘𝐺) ⊆ (Trails‘𝐺) * Simple paths are paths (spthispth 30189): (SPaths‘𝐺) ⊆ (Paths‘𝐺) * Closed walks are walks (clwlkiswlk 30241): (ClWalks‘𝐺) ⊆ (Walks‘𝐺) * Circuits are closed walks (crctisclwlk 30261): (Circuits‘𝐺) ⊆ (ClWalks‘𝐺) * Circuits are trails (crctistrl 30262): (Circuits‘𝐺) ⊆ (Trails‘𝐺) * Cycles are paths (cyclispth 30264): (Cycles‘𝐺) ⊆ (Paths‘𝐺) * Cycles are circuits (cycliscrct 30267): (Cycles‘𝐺) ⊆ (Circuits‘𝐺) * (Non-trivial) cycles are not simple paths (cyclnspth 30269): (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) * Eulerian paths are trails (eupthistrl 30692): (EulerPaths‘𝐺) ⊆ (Trails‘𝐺) Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e., omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 14581, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definition df-wwlks 30299 for WWalks, and the representation of a walk as sequence of its vertices is called "walk as word". Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In this case, the general definitions of walks and the definition of walks as words are equivalent, see wlkiswwlks 30345. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph). Based on this definition of WWalks, the class of all walks as word, more restrictive constructs are defined analogously to the general definition of a walk: * WWalksN (walks of length N as word, df-wwlksn 30300): n = N * WSPathsN (simple paths of length N as word, df-wspthsn 30302): p(i) =/= p(j) if i =/= j and n = N * ClWWalks (closed walks as word, df-clwwlk 30453): p(0) = p(n) * ClWWalksN (closed walks of length N as word, df-clwwlkn 30496): p(0) = p(n) and n = N Finally, there are a couple of definitions for (special) walks ⟨𝐹, 𝑃⟩ having fixed endpoints 𝐴 and 𝐵: * Walks with particular endpoints (df-wlkson 30061): 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 * Trails with particular endpoints (df-trlson 30156): 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 * Paths with particular endpoints (df-pthson 30181): 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 * Simple paths with particular endpoints (df-spthson 30182): 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 * Walks of a fixed length 𝑁 as words with particular endpoints (df-wwlksnon 30301): (𝐴(𝑁 WWalksNOn 𝐺)𝐵) * Simple paths of a fixed length 𝑁 as words with particular endpoints (df-wspthsnon 30303): (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) * Closed Walks of a fixed length 𝑁 as words anchored at a particular vertex 𝐴 (df-wwlksnon 30301): (𝐴(ClWWalksNOn‘𝐺)𝑁)
17.3.1  Walks A "walk" within a graph is usually defined for simple graphs, multigraphs or even pseudographs as "alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. This definition requires the edges to connect two vertices at most (loops are also allowed: if e(i) is a loop, then x(i-1) = x(i)). For hypergraphs containing hyperedges (i.e. edges connecting more than two vertices), however, a more general definition is needed. Two approaches for a definition applicable for arbitrary hypergraphs are used in the literature: "walks on the vertex level" and "walks on the edge level" (see Aksoy, Joslyn, Marrero, Praggastis, Purvine: "Hypernetwork science via high-order hypergraph walks", June 2020, https://doi.org/10.1140/epjds/s13688-020-00231-0): "walks on the edge level": For a positive integer s, an s-walk of length k between hyperedges f and g is a sequence of hyperedges, f=e(0), e(1), ... , e(k)=g, where for j=1, ... , k, e(j-1) =/= e(j) and e(j-1) and e(j) have at least s vertices in common (according to Aksoy et al.). "walks on the vertex level": For a positive integer s, an s-walk of length k between vertices a and b is a sequence of vertices, a=v(0), v(1), ... , v(k)=b, where for j=1, ... , k, v(j-1) and v(j) are connected by at least s edges (analogous to Aksoy et al.). There are two imperfections for the definition for "walks on the edge level": one is that a walk of length 1 consists of two edges (or a walk of length 0 consists of one edge), whereas a walk of length 1 on the vertex level consists of two vertices and one edge (or a walk of length 0 consists of one vertex and no edge). The other is that edges, especially loops, can be traversed only once (and not repeatedly) because of the condition e(j-1) =/= e(j). The latter is avoided in the definition for EdgWalks, see df-ewlks 30059. To be compatible with the (usual) definition of walks for pseudographs, walks also suitable for arbitrary hypergraphs are defined "on the vertex level" in the following as Walks, see df-wlks 30060, restricting s to 1. wlk1ewlk 30100 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".
Syntax	cewlks 30056	Extend class notation with s-walks "on the edge level" (of a hypergraph).
class EdgWalks
Syntax	cwlks 30057	Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks
Syntax	cwlkson 30058	Extend class notation with walks between two vertices (within a graph).
class WalksOn
Definition	df-ewlks 30059*	Define the set of all s-walks of edges (in a hypergraph) corresponding to s-walks "on the edge level" discussed in Aksoy et al. For an extended nonnegative integer s, an s-walk is a sequence of hyperedges, e(0), e(1), ... , e(k), where e(j-1) and e(j) have at least s vertices in common (for j=1, ... , k). In contrast to the definition in Aksoy et al., 𝑠 = 0 (a 0-walk is an arbitrary sequence of hyperedges) and 𝑠 = +∞ (then the number of common vertices of two adjacent hyperedges must be infinite) are allowed. Furthermore, it is not forbidden that adjacent hyperedges are equal. (Contributed by AV, 4-Jan-2021.)
⊢ EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓 ∣ [(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑠 ≤ (♯‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓‘𝑘)))))})
Definition	df-wlks 30060*	Define the set of all walks (in a hypergraph). Such walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see wlk1walk 30099) discussed in Aksoy et al. The predicate 𝐹(Walks‘𝐺)𝑃 can be read as "The pair ⟨𝐹, 𝑃⟩ represents a walk in a graph 𝐺", see also iswlk 30071. The condition {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝‘𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝‘𝑘). According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the 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). (Contributed by AV, 30-Dec-2020.)
⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})
Definition	df-wlkson 30061*	Define the collection of walks with particular endpoints (in a hypergraph). The predicate 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 can be read as "The pair ⟨𝐹, 𝑃⟩ represents a walk from vertex 𝐴 to vertex 𝐵 in a graph 𝐺", see also iswlkon 30116. This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
⊢ WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
Theorem	ewlksfval 30062*	The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓‘𝑘)))))})
Theorem	isewlk 30063*	Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))))
Theorem	ewlkprop 30064*	Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘))))))
Theorem	ewlkinedg 30065	The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(♯‘𝐹))) → 𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹‘𝐾)))))
Theorem	ewlkle 30066	An s-walk of edges is also a t-walk of edges if 𝑡 ≤ 𝑠. (Contributed by AV, 4-Jan-2021.)
⊢ ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ0* ∧ 𝑇 ≤ 𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))
Theorem	upgrewlkle2 30067	In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (♯‘𝐹)) → 𝑆 ≤ 2)
Theorem	wkslem1 30068	Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)
⊢ (𝐴 = 𝐵 → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹‘𝐵)))))
Theorem	wkslem2 30069	Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.)
⊢ ((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃‘𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹‘𝐴)) = {(𝑃‘𝐴)}, {(𝑃‘𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹‘𝐴))) ↔ if-((𝑃‘𝐵) = (𝑃‘𝐶), (𝐼‘(𝐹‘𝐵)) = {(𝑃‘𝐵)}, {(𝑃‘𝐵), (𝑃‘𝐶)} ⊆ (𝐼‘(𝐹‘𝐵)))))
Theorem	wksfval 30070*	The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ 𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓‘𝑘))))})
Theorem	iswlk 30071*	Properties of a pair of functions to be/represent a walk. (Contributed by AV, 30-Dec-2020.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
Theorem	wlkprop 30072*	Properties of a walk. (Contributed by AV, 5-Nov-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
Theorem	wlkv 30073	The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
Theorem	iswlkg 30074*	Generalization of iswlk 30071: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
Theorem	wlkf 30075	The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
Theorem	wlkcl 30076	A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 30691. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
Theorem	wlkp 30077	The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶𝑉)
Theorem	wlkpwrd 30078	The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ∈ Word 𝑉)
Theorem	wlklenvp1 30079	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.) (Revised by AV, 1-May-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))
Theorem	wksv 30080*	The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.)
⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
Theorem	wlkn0 30081	The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃 ≠ ∅)
Theorem	wlklenvm1 30082	The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1))
Theorem	ifpsnprss 30083	Lemma for wlkvtxeledg 30084: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)
⊢ (if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)
Theorem	wlkvtxeledg 30084*	Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
Theorem	wlkvtxiedg 30085*	The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)
Theorem	relwlk 30086	The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)
⊢ Rel (Walks‘𝐺)
Theorem	wlkvv 30087	If there is at least one walk in the graph, all walks are in the universal class of ordered pairs. (Contributed by AV, 2-Jan-2021.)
⊢ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) → 𝑊 ∈ (V × V))
Theorem	wlkop 30088	A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)
⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
Theorem	wlkcpr 30089	A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
⊢ (𝑊 ∈ (Walks‘𝐺) ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
Theorem	wlk2f 30090*	If there is a walk 𝑊 there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
⊢ (𝑊 ∈ (Walks‘𝐺) → ∃𝑓∃𝑝 𝑓(Walks‘𝐺)𝑝)
Theorem	wlkcomp 30091*	A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ ((𝐺 ∈ 𝑈 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))))
Theorem	wlkcompim 30092*	Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))))
Theorem	wlkelwrd 30093	The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ (𝑊 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉))
Theorem	wlkeq 30094*	Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)
⊢ ((𝐴 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st ‘𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st ‘𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st ‘𝐴)‘𝑥) = ((1st ‘𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd ‘𝐴)‘𝑥) = ((2nd ‘𝐵)‘𝑥))))
Theorem	edginwlk 30095	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((Fun 𝐼 ∧ 𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸)
Theorem	upgredginwlk 30096	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(♯‘𝐹)) → (𝐼‘(𝐹‘𝐾)) ∈ 𝐸))
Theorem	iedginwlk 30097	The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((Fun 𝐼 ∧ 𝐹(Walks‘𝐺)𝑃 ∧ 𝑋 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹‘𝑋)) ∈ ran 𝐼)
Theorem	wlkl1loop 30098	A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
⊢ (((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))
Theorem	wlk1walk 30099*	A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹‘𝑘)))))
Theorem	wlk1ewlk 30100	A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ (𝐺 EdgWalks 1))