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Theorem List for Metamath Proof Explorer - 27901-28000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremumgr2v2e 27901 A multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → 𝐺 ∈ UMGraph)
 
Theoremumgr2v2enb1 27902 In a multigraph with two edges connecting the same two vertices, each of the vertices has one neighbor. (Contributed by AV, 18-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐺 NeighbVtx 𝐴) = {𝐵})
 
Theoremumgr2v2evd2 27903 In a multigraph with two edges connecting the same two vertices, each of the vertices has degree 2. (Contributed by AV, 18-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((VtxDeg‘𝐺)‘𝐴) = 2)
 
Theoremhashnbusgrvd 27904 In a simple graph, the number of neighbors of a vertex is the degree of this vertex. This theorem does not hold for (simple) pseudographs, because a vertex connected with itself only by a loop has no neighbors, see uspgrloopnb0 27895, but degree 2, see uspgrloopvd2 27896. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 27902, but also degree 2, see umgr2v2evd2 27903. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (♯‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))
 
Theoremusgruvtxvdb 27905 In a finite simple graph with n vertices a vertex is universal iff the vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (𝑈 ∈ (UnivVtx‘𝐺) ↔ ((VtxDeg‘𝐺)‘𝑈) = ((♯‘𝑉) − 1)))
 
Theoremvdiscusgrb 27906* A finite simple graph with n vertices is complete iff every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1)))
 
Theoremvdiscusgr 27907* In a finite complete simple graph with n vertices every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((♯‘𝑉) − 1) → 𝐺 ∈ ComplUSGraph))
 
Theoremvtxdusgradjvtx 27908* The degree of a vertex in a simple graph is the number of vertices adjacent to this vertex. (Contributed by Alexander van der Vekens, 9-Jul-2018.) (Revised by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = (♯‘{𝑣𝑉 ∣ {𝑈, 𝑣} ∈ 𝐸}))
 
Theoremusgrvd0nedg 27909* 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, 20-Dec-2017.) (Revised by AV, 16-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (((VtxDeg‘𝐺)‘𝑈) = 0 → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ 𝐸))
 
Theoremuhgrvd00 27910* If every vertex in a hypergraph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UHGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))
 
Theoremusgrvd00 27911* 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.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))
 
Theoremvdegp1ai 27912* 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 𝑋𝑈𝑌, 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)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   𝑌𝑉    &   𝑌𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = 𝑃
 
Theoremvdegp1bi 27913* 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 𝑋𝑈, 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)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑈, 𝑋}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
 
Theoremvdegp1ci 27914* 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 𝑋𝑈, 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)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑈}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)
 
Theoremvtxdginducedm1lem1 27915 Lemma 1 for vtxdginducedm1 27919: 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       (iEdg‘𝑆) = 𝑃
 
Theoremvtxdginducedm1lem2 27916* Lemma 2 for vtxdginducedm1 27919: 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       dom (iEdg‘𝑆) = 𝐼
 
Theoremvtxdginducedm1lem3 27917* Lemma 3 for vtxdginducedm1 27919: 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))
 
Theoremvtxdginducedm1lem4 27918* Lemma 4 for vtxdginducedm1 27919. (Contributed by AV, 17-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (𝑊 ∈ (𝑉 ∖ {𝑁}) → (♯‘{𝑘𝐽 ∣ (𝐸𝑘) = {𝑊}}) = 0)
 
Theoremvtxdginducedm1 27919* 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (♯‘{𝑙𝐽𝑣 ∈ (𝐸𝑙)}))
 
Theoremvtxdginducedm1fi 27920* 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (♯‘{𝑙𝐽𝑣 ∈ (𝐸𝑙)})))
 
Theoremfinsumvtxdg2ssteplem1 27921* Lemma for finsumvtxdg2sstep 27925. (Contributed by AV, 15-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (♯‘𝐸) = ((♯‘𝑃) + (♯‘𝐽)))
 
Theoremfinsumvtxdg2ssteplem2 27922* Lemma for finsumvtxdg2sstep 27925. (Contributed by AV, 12-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))
 
Theoremfinsumvtxdg2ssteplem3 27923* Lemma for finsumvtxdg2sstep 27925. (Contributed by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(♯‘{𝑖𝐽𝑣 ∈ (𝐸𝑖)}) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (♯‘𝐽))
 
Theoremfinsumvtxdg2ssteplem4 27924* Lemma for finsumvtxdg2sstep 27925. (Contributed by AV, 12-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((♯‘𝐽) + (♯‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))) = (2 · ((♯‘𝑃) + (♯‘𝐽))))
 
Theoremfinsumvtxdg2sstep 27925* Induction step of finsumvtxdg2size 27926: 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 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((𝑃 ∈ Fin → Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (♯‘𝑃))) → Σ𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = (2 · (♯‘𝐸))))
 
Theoremfinsumvtxdg2size 27926* 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 27927) 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) → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))
 
Theoremfusgr1th 27927* 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 → Σ𝑣𝑉 (𝐷𝑣) = (2 · (♯‘𝐼)))
 
Theoremfinsumvtxdgeven 27928* 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 ∥ Σ𝑣𝑉 (𝐷𝑣))
 
Theoremvtxdgoddnumeven 27929* 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 ∥ (♯‘{𝑣𝑉 ∣ ¬ 2 ∥ (𝐷𝑣)}))
 
Theoremfusgrvtxdgonume 27930* 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 ∥ (♯‘{𝑣𝑉 ∣ ¬ 2 ∥ (𝐷𝑣)}))
 
16.2.11  Regular graphs

With df-rgr 27933 and df-rusgr 27934, 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 27970), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 27968). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).

 
Syntaxcrgr 27931 Extend class notation to include the class of all regular graphs.
class RegGraph
 
Syntaxcrusgr 27932 Extend class notation to include the class of all regular simple graphs.
class RegUSGraph
 
Definitiondf-rgr 27933* 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)}
 
Definitiondf-rusgr 27934* 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)}
 
Theoremisrgr 27935* 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)       ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
 
Theoremrgrprop 27936* 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* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
 
Theoremisrusgr 27937 The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.)
((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))
 
Theoremrusgrprop 27938 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))
 
Theoremrusgrrgr 27939 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 𝐾𝐺 RegGraph 𝐾)
 
Theoremrusgrusgr 27940 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 𝐾𝐺 ∈ USGraph)
 
Theoremfinrusgrfusgr 27941 A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph)
 
Theoremisrusgr0 27942* 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)       ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))
 
Theoremrusgrprop0 27943* 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* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))
 
Theoremusgreqdrusgr 27944* 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* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)
 
Theoremfusgrregdegfi 27945* 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 ≠ ∅) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾𝐾 ∈ ℕ0))
 
Theoremfusgrn0eqdrusgr 27946* 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 ≠ ∅) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾𝐺 RegUSGraph 𝐾))
 
Theoremfrusgrnn0 27947 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 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)
 
Theorem0edg0rgr 27948 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.)
((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0)
 
Theoremuhgr0edg0rgr 27949 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0)
 
Theoremuhgr0edg0rgrb 27950 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) = ∅))
 
Theoremusgr0edg0rusgr 27951 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) = ∅))
 
Theorem0vtxrgr 27952* 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘)
 
Theorem0vtxrusgr 27953* 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.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)
 
Theorem0uhgrrusgr 27954* 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)
 
Theorem0grrusgr 27955 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
 
Theorem0grrgr 27956 The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
𝑘 ∈ ℕ0* ∅ RegGraph 𝑘
 
Theoremcusgrrusgr 27957 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))
 
Theoremcusgrm1rusgr 27958 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)))
 
Theoremrusgrpropnb 27959* 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* ∧ ∀𝑣𝑉 (♯‘(𝐺 NeighbVtx 𝑣)) = 𝐾))
 
Theoremrusgrpropedg 27960* 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* ∧ ∀𝑣𝑉 (♯‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣𝑒}) = 𝐾))
 
Theoremrusgrpropadjvtx 27961* 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* ∧ ∀𝑣𝑉 (♯‘{𝑘𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))
 
Theoremrusgrnumwrdl2 27962* 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 𝐾𝑃𝑉) → (♯‘{𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)
 
Theoremrusgr1vtxlem 27963* Lemma for rusgr1vtx 27964. (Contributed by AV, 27-Dec-2020.)
(((∀𝑣𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣𝑉 𝐴 = ∅) ∧ (𝑉𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0)
 
Theoremrusgr1vtx 27964 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)
 
Theoremrgrusgrprc 27965* 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
 
Theoremrusgrprc 27966 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegUSGraph 0} ∉ V
 
Theoremrgrprc 27967 The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegGraph 0} ∉ V
 
Theoremrgrprcx 27968* 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
 
Theoremrgrx0ndm 27969* 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
 
Theoremrgrx0nd 27970* 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 27975. 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 ≤ 𝑘𝑛, (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 28069): 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 28093): 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 28094): 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 28148): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n).

* Circuits (df-crcts 28163): "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 28164): 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 28571): 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 3926 and elopabran 5476 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 28151, see clwlkwlk 28152 and clwlkswks 28153):

* Trails are walks (trliswlk 28074): (Trails‘𝐺) ⊆ (Walks‘𝐺)

* Paths are trails (pthistrl 28102): (Paths‘𝐺) ⊆ (Trails‘𝐺)

* Simple paths are paths (spthispth 28103): (SPaths‘𝐺) ⊆ (Paths‘𝐺)

* Closed walks are walks (clwlkiswlk 28151): (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

* Circuits are closed walks (crctisclwlk 28171): (Circuits‘𝐺) ⊆ (ClWalks‘𝐺)

* Circuits are trails (crctistrl 28172): (Circuits‘𝐺) ⊆ (Trails‘𝐺)

* Cycles are paths (cyclispth 28174): (Cycles‘𝐺) ⊆ (Paths‘𝐺)

* Cycles are circuits (cycliscrct 28176): (Cycles‘𝐺) ⊆ (Circuits‘𝐺)

* (Non-trivial) cycles are not simple paths (cyclnspth 28177): (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃))

* Eulerian paths are trails (eupthistrl 28584): (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 14227, 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 28204 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 28250. 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 28205): n = N

* WSPathsN (simple paths of length N as word, df-wspthsn 28207): p(i) =/= p(j) if i =/= j and n = N

* ClWWalks (closed walks as word, df-clwwlk 28355): p(0) = p(n)

* ClWWalksN (closed walks of length N as word, df-clwwlkn 28398): 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 27976): 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃

* Trails with particular endpoints (df-trlson 28070): 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃

* Paths with particular endpoints (df-pthson 28095): 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃

* Simple paths with particular endpoints (df-spthson 28096): 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃

* Walks of a fixed length 𝑁 as words with particular endpoints (df-wwlksnon 28206): (𝐴(𝑁 WWalksNOn 𝐺)𝐵)

* Simple paths of a fixed length 𝑁 as words with particular endpoints (df-wspthsnon 28208): (𝐴(𝑁 WSPathsNOn 𝐺)𝐵)

* Closed Walks of a fixed length 𝑁 as words anchored at a particular vertex 𝐴 (df-wwlksnon 28206): (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 27974. 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 27975, restricting s to 1. wlk1ewlk 28016 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".

 
Syntaxcewlks 27971 Extend class notation with s-walks "on the edge level" (of a hypergraph).
class EdgWalks
 
Syntaxcwlks 27972 Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks
 
Syntaxcwlkson 27973 Extend class notation with walks between two vertices (within a graph).
class WalksOn
 
Definitiondf-ewlks 27974* 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‘(𝑓𝑘)))))})
 
Definitiondf-wlks 27975* 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 28015) 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 27986.

The condition {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝𝑘).

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-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
 
Definitiondf-wlkson 27976* 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 28034. 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)}))
 
Theoremewlksfval 27977* The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝑓))𝑆 ≤ (♯‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))})
 
Theoremisewlk 27978* Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝑆 ∈ ℕ0*𝐹𝑈) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(♯‘𝐹))𝑆 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹𝑘)))))))
 
Theoremewlkprop 27979* 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‘(𝐹𝑘))))))
 
Theoremewlkinedg 27980 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‘(𝐹𝐾)))))
 
Theoremewlkle 27981 An s-walk of edges is also a t-walk of edges if 𝑡𝑠. (Contributed by AV, 4-Jan-2021.)
((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ0*𝑇𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))
 
Theoremupgrewlkle2 27982 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)
 
Theoremwkslem1 27983 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-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵)))))
 
Theoremwkslem2 27984 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-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃𝐶), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵)))))
 
Theoremwksfval 27985* The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))})
 
Theoremiswlk 27986* Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
 
Theoremwlkprop 27987* Properties of a walk. (Contributed by AV, 5-Nov-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
 
Theoremwlkv 27988 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))
 
Theoremiswlkg 27989* Generalization of iswlk 27986: 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)       (𝐺𝑊 → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
 
Theoremwlkf 27990 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)𝑃𝐹 ∈ Word dom 𝐼)
 
Theoremwlkcl 27991 A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 28583. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
(𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)
 
Theoremwlkp 27992 The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃𝑃:(0...(♯‘𝐹))⟶𝑉)
 
Theoremwlkpwrd 27993 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)𝑃𝑃 ∈ Word 𝑉)
 
Theoremwlklenvp1 27994 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))
 
Theoremwksv 27995* 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
 
TheoremwksvOLD 27996* Obsolete version of wksv 27995 as of 11-Dec-2024. (Contributed by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
{⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V
 
Theoremwlkn0 27997 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)𝑃𝑃 ≠ ∅)
 
Theoremwlklenvm1 27998 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))
 
Theoremifpsnprss 27999 Lemma for wlkvtxeledg 28000: 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)
 
Theoremwlkvtxeledg 28000* 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)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
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