P. 271 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 27001-27100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	vdegp1ai 27001*	The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋 ≠ 𝑈 ≠ 𝑌, yields degree 𝑃 as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑌 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = 𝑃
Theorem	vdegp1bi 27002*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑈, 𝑋}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
Theorem	vdegp1ci 27003*	The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑈} to the edge set, where 𝑋 ≠ 𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝑈 ∈ 𝑉    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ⊢ ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   ⊢ (Vtx‘𝐹) = 𝑉    &   ⊢ 𝑋 ∈ 𝑉    &   ⊢ 𝑋 ≠ 𝑈    &   ⊢ (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑈}”⟩)    ⇒   ⊢ ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
Theorem	vtxdginducedm1lem1 27004	Lemma 1 for vtxdginducedm1 27008: 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⟩    ⇒   ⊢ (iEdg‘𝑆) = 𝑃
Theorem	vtxdginducedm1lem2 27005*	Lemma 2 for vtxdginducedm1 27008: 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 27006*	Lemma 3 for vtxdginducedm1 27008: 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 27007*	Lemma 4 for vtxdginducedm1 27008. (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 27008*	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 27009*	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 27010*	Lemma for finsumvtxdg2sstep 27014. (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 27011*	Lemma for finsumvtxdg2sstep 27014. (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 27012*	Lemma for finsumvtxdg2sstep 27014. (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 27013*	Lemma for finsumvtxdg2sstep 27014. (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 27014*	Induction step of finsumvtxdg2size 27015: 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 27015*	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 27016) 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 27016*	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 27017*	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 27018*	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 27019*	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)}))
16.2.11  Regular graphs With df-rgr 27022 and df-rusgr 27023, 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 27059), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 27057). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).
Syntax	crgr 27020	Extend class notation to include the class of all regular graphs.
class RegGraph
Syntax	crusgr 27021	Extend class notation to include the class of all regular simple graphs.
class RegUSGraph
Definition	df-rgr 27022*	Define the "k-regular" predicate, which is true for a "graph" being k-regular: read 𝐺RegGraph𝐾 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 27023*	Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read 𝐺RegUSGraph𝐾 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 ∧ 𝑔RegGraph𝑘)}
Theorem	isrgr 27024*	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) → (𝐺RegGraph𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
Theorem	rgrprop 27025*	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)    ⇒   ⊢ (𝐺RegGraph𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
Theorem	isrusgr 27026	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) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾)))
Theorem	rusgrprop 27027	The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐺RegGraph𝐾))
Theorem	rusgrrgr 27028	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.)
⊢ (𝐺RegUSGraph𝐾 → 𝐺RegGraph𝐾)
Theorem	rusgrusgr 27029	A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
⊢ (𝐺RegUSGraph𝐾 → 𝐺 ∈ USGraph)
Theorem	finrusgrfusgr 27030	A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
Theorem	isrusgr0 27031*	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) → (𝐺RegUSGraph𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾)))
Theorem	rusgrprop0 27032*	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)    ⇒   ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾))
Theorem	usgreqdrusgr 27033*	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) → 𝐺RegUSGraph𝐾)
Theorem	fusgrregdegfi 27034*	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 27035*	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 → 𝐺RegUSGraph𝐾))
Theorem	frusgrnn0 27036	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 ∧ 𝐺RegUSGraph𝐾 ∧ 𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
Theorem	0edg0rgr 27037	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) = ∅) → 𝐺RegGraph0)
Theorem	uhgr0edg0rgr 27038	A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
⊢ ((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺RegGraph0)
Theorem	uhgr0edg0rgrb 27039	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 → (𝐺RegGraph0 ↔ (Edg‘𝐺) = ∅))
Theorem	usgr0edg0rusgr 27040	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 → (𝐺RegUSGraph0 ↔ (Edg‘𝐺) = ∅))
Theorem	0vtxrgr 27041*	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* 𝐺RegGraph𝑘)
Theorem	0vtxrusgr 27042*	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* 𝐺RegUSGraph𝑘)
Theorem	0uhgrrusgr 27043*	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* 𝐺RegUSGraph𝑘)
Theorem	0grrusgr 27044	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* ∅RegUSGraph𝑘
Theorem	0grrgr 27045	The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
⊢ ∀𝑘 ∈ ℕ0* ∅RegGraph𝑘
Theorem	cusgrrusgr 27046	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 ≠ ∅) → 𝐺RegUSGraph((♯‘𝑉) − 1))
Theorem	cusgrm1rusgr 27047	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 ↔ 𝐺RegUSGraph((♯‘𝑉) − 1)))
Theorem	rusgrpropnb 27048*	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)    ⇒   ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
Theorem	rusgrpropedg 27049*	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)    ⇒   ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣 ∈ 𝑒}) = 𝐾))
Theorem	rusgrpropadjvtx 27050*	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)    ⇒   ⊢ (𝐺RegUSGraph𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣 ∈ 𝑉 (♯‘{𝑘 ∈ 𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
Theorem	rusgrnumwrdl2 27051*	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)    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑃 ∈ 𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
Theorem	rusgr1vtxlem 27052*	Lemma for rusgr1vtx 27053. (Contributed by AV, 27-Dec-2020.)
⊢ (((∀𝑣 ∈ 𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣 ∈ 𝑉 𝐴 = ∅) ∧ (𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0)
Theorem	rusgr1vtx 27053	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 ∧ 𝐺RegUSGraph𝐾) → 𝐾 = 0)
Theorem	rgrusgrprc 27054*	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 27055	The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∣ 𝑔RegUSGraph0} ∉ V
Theorem	rgrprc 27056	The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
⊢ {𝑔 ∣ 𝑔RegGraph0} ∉ V
Theorem	rgrprcx 27057*	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 27058*	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 27059*	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) = ∅
16.3  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 27064. 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 27159): 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 27184): 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 27185): 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 27239): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n). * Circuits (df-crcts 27254): "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 27255): 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 27664): 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 3893 and elopabran 5336 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 27242, see clwlkwlk 27243 and clwlkswks 27244): * Trails are walks (trliswlk 27164): (Trails‘𝐺) ⊆ (Walks‘𝐺) * Paths are trails (pthistrl 27193): (Paths‘𝐺) ⊆ (Trails‘𝐺) * Simple paths are paths (spthispth 27194): (SPaths‘𝐺) ⊆ (Paths‘𝐺) * Closed walks are walks (clwlkiswlk 27242): (ClWalks‘𝐺) ⊆ (Walks‘𝐺) * Circuits are closed walks (crctisclwlk 27262): (Circuits‘𝐺) ⊆ (ClWalks‘𝐺) * Circuits are trails (crctistrl 27263): (Circuits‘𝐺) ⊆ (Trails‘𝐺) * Cycles are paths (cyclispth 27265): (Cycles‘𝐺) ⊆ (Paths‘𝐺) * Cycles are circuits (cycliscrct 27267): (Cycles‘𝐺) ⊆ (Circuits‘𝐺) * (Non-trivial) cycles are not simple paths (cyclnspth 27268): (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃)) * Eulerian paths are trails (eupthistrl 27677): (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 13708, 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 27295 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 27341. 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 27296): n = N * WSPathsN (simple paths of length N as word, df-wspthsn 27298): p(i) =/= p(j) if i =/= j and n = N * ClWWalks (closed walks as word, df-clwwlk 27447): p(0) = p(n) * ClWWalksN (closed walks of length N as word, df-clwwlkn 27490): 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 27065): 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 * Trails with particular endpoints (df-trlson 27160): 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 * Paths with particular endpoints (df-pthson 27186): 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 * Simple paths with particular endpoints (df-spthson 27187): 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 * Walks of a fixed length 𝑁 as words with particular endpoints (df-wwlksnon 27297): (𝐴(𝑁 WWalksNOn 𝐺)𝐵) * Simple paths of a fixed length 𝑁 as words with particular endpoints (df-wspthsnon 27299): (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) * Closed Walks of a fixed length 𝑁 as words anchored at a particular vertex 𝐴 (df-wwlksnon 27297): (𝐴(ClWWalksNOn‘𝐺)𝑁)
16.3.1  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 27063. 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 27064, restricting s to 1. wlk1ewlk 27104 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".
Syntax	cewlks 27060	Extend class notation with s-walks "on the edge level" (of a hypergraph).
class EdgWalks
Syntax	cwlks 27061	Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks
Syntax	cwlkson 27062	Extend class notation with walks between two vertices (within a graph).
class WalksOn
Definition	df-ewlks 27063*	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 27064*	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 27103) 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 27075. 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 27065*	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 27121. 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 27066*	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 27067*	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 27068*	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 27069	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 27070	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 27071	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 27072	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 27073	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 27074*	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 27075*	Properties of a pair of functions to be 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 27076*	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 27077	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 27078*	Generalization of iswlk 27075: 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 27079	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 27080	A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 27676. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
Theorem	wlkp 27081	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 27082	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 27083	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 27084*	The class of walks is a set. (Contributed by AV, 15-Jan-2021.)
⊢ {⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
Theorem	wlkn0 27085	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 27086	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 27087	Lemma for wlkvtxeledg 27088: 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 27088*	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 27089*	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 27090	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 27091	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 27092	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 27093	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 27094*	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 27095*	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 27096*	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 27097	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 27098*	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 27099	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 27100	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))