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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem1loopgredg 27301 The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑 → (Edg‘𝐺) = {{𝑁}})

Theorem1loopgrnb0 27302 In a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑 → (𝐺 NeighbVtx 𝑁) = ∅)

Theorem1loopgrvd2 27303 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝑁) = 2)

Theorem1loopgrvd0 27304 The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})    &   (𝜑𝐾 ∈ (𝑉 ∖ {𝑁}))       (𝜑 → ((VtxDeg‘𝐺)‘𝐾) = 0)

Theorem1hevtxdg0 27305 The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 is 0 if 𝐷 is not incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.)
(𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐷𝑉)    &   (𝜑𝐸𝑌)    &   (𝜑𝐷𝐸)       (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 0)

Theorem1hevtxdg1 27306 The vertex degree of vertex 𝐷 in a graph 𝐺 with only one hyperedge 𝐸 (not being a loop) is 1 if 𝐷 is incident with the edge 𝐸. (Contributed by AV, 2-Mar-2021.) (Proof shortened by AV, 17-Apr-2021.)
(𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐷𝑉)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑𝐷𝐸)    &   (𝜑 → 2 ≤ (♯‘𝐸))       (𝜑 → ((VtxDeg‘𝐺)‘𝐷) = 1)

Theorem1hegrvtxdg1 27307 The vertex degree of a graph with one hyperedge, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   (𝜑 → (Vtx‘𝐺) = 𝑉)       (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)

Theorem1hegrvtxdg1r 27308 The vertex degree of a graph with one hyperedge, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 23-Feb-2021.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)    &   (𝜑 → (Vtx‘𝐺) = 𝑉)       (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)

Theorem1egrvtxdg1 27309 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝐵) = 1)

Theorem1egrvtxdg1r 27310 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)

Theorem1egrvtxdg0 27311 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐷𝑉)    &   (𝜑𝐶𝐷)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐷}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)

Theoremp1evtxdeqlem 27312 Lemma for p1evtxdeq 27313 and p1evtxdp1 27314. (Contributed by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸𝑌)       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)‘𝑈)))

Theoremp1evtxdeq 27313 If an edge 𝐸 which does not contain vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸𝑌)    &   (𝜑𝑈𝐸)       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))

Theoremp1evtxdp1 27314 If an edge 𝐸 (not being a loop) which contains vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is increased by 1. (Contributed by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑𝑈𝐸)    &   (𝜑 → 2 ≤ (♯‘𝐸))       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1))

Theoremuspgrloopvtx 27315 The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)

Theoremuspgrloopvtxel 27316 A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuspgrloopiedg 27317 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Theoremuspgrloopedg 27318 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Theoremuspgrloopnb0 27319 In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049), the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = ∅)

Theoremuspgrloopvd2 27320 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27049), the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = 2)

Theoremumgr2v2evtx 27321 The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)

Theoremumgr2v2evtxel 27322 A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       ((𝑉𝑊𝐴𝑉) → 𝐴 ∈ (Vtx‘𝐺))

Theoremumgr2v2eiedg 27323 The edge function in 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) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})

Theoremumgr2v2eedg 27324 The set of edges in 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) → (Edg‘𝐺) = {{𝐴, 𝐵}})

Theoremumgr2v2e 27325 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 27326 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 27327 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 27328 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 27319, but degree 2, see uspgrloopvd2 27320. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 27326, but also degree 2, see umgr2v2evd2 27327. (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 27329 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 27330* 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 27331* 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 27332* 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 27333* 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 27334* 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 27335* 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 27336* 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 27337* 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 27338* 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 27339 Lemma 1 for vtxdginducedm1 27343: 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 27340* Lemma 2 for vtxdginducedm1 27343: 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 27341* Lemma 3 for vtxdginducedm1 27343: 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 27342* Lemma 4 for vtxdginducedm1 27343. (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 27343* 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 27344* 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 27345* Lemma for finsumvtxdg2sstep 27349. (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 27346* Lemma for finsumvtxdg2sstep 27349. (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 27347* Lemma for finsumvtxdg2sstep 27349. (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 27348* Lemma for finsumvtxdg2sstep 27349. (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 27349* Induction step of finsumvtxdg2size 27350: 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 27350* 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 27351) 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 27351* 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 27352* 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 27353* 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 27354* 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 27357 and df-rusgr 27358, 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 27394), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 27392). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).

Syntaxcrgr 27355 Extend class notation to include the class of all regular graphs.
class RegGraph

Syntaxcrusgr 27356 Extend class notation to include the class of all regular simple graphs.
class RegUSGraph

Definitiondf-rgr 27357* 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 27358* 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 27359* 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 27360* 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 27361 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 27362 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 27363 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 27364 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 27365 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 27366* 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 27367* 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 27368* 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 27369* 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 27370* 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 27371 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 27372 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 27373 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0)

Theoremuhgr0edg0rgrb 27374 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 27375 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 27376* 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 27377* 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 27378* 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 27379 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 27380 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 27381 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 27382 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 27383* 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 27384* 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 27385* 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 27386* 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 27387* Lemma for rusgr1vtx 27388. (Contributed by AV, 27-Dec-2020.)
(((∀𝑣𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣𝑉 𝐴 = ∅) ∧ (𝑉𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0)

Theoremrusgr1vtx 27388 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 27389* 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 27390 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegUSGraph 0} ∉ V

Theoremrgrprc 27391 The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegGraph 0} ∉ V

Theoremrgrprcx 27392* 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 27393* 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 27394* 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 27399. 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 27492): 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 27515): 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 27516): 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 27570): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n).

* Circuits (df-crcts 27585): "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 27586): 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 27993): 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 3919 and elopabran 5414 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 27573, see clwlkwlk 27574 and clwlkswks 27575):

* Trails are walks (trliswlk 27497): (Trails‘𝐺) ⊆ (Walks‘𝐺)

* Paths are trails (pthistrl 27524): (Paths‘𝐺) ⊆ (Trails‘𝐺)

* Simple paths are paths (spthispth 27525): (SPaths‘𝐺) ⊆ (Paths‘𝐺)

* Closed walks are walks (clwlkiswlk 27573): (ClWalks‘𝐺) ⊆ (Walks‘𝐺)

* Circuits are closed walks (crctisclwlk 27593): (Circuits‘𝐺) ⊆ (ClWalks‘𝐺)

* Circuits are trails (crctistrl 27594): (Circuits‘𝐺) ⊆ (Trails‘𝐺)

* Cycles are paths (cyclispth 27596): (Cycles‘𝐺) ⊆ (Paths‘𝐺)

* Cycles are circuits (cycliscrct 27598): (Cycles‘𝐺) ⊆ (Circuits‘𝐺)

* (Non-trivial) cycles are not simple paths (cyclnspth 27599): (𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃))

* Eulerian paths are trails (eupthistrl 28006): (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 13861, 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 27626 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 27672. 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 27627): n = N

* WSPathsN (simple paths of length N as word, df-wspthsn 27629): p(i) =/= p(j) if i =/= j and n = N

* ClWWalks (closed walks as word, df-clwwlk 27777): p(0) = p(n)

* ClWWalksN (closed walks of length N as word, df-clwwlkn 27820): 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 27400): 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃

* Trails with particular endpoints (df-trlson 27493): 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃

* Paths with particular endpoints (df-pthson 27517): 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃

* Simple paths with particular endpoints (df-spthson 27518): 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃

* Walks of a fixed length 𝑁 as words with particular endpoints (df-wwlksnon 27628): (𝐴(𝑁 WWalksNOn 𝐺)𝐵)

* Simple paths of a fixed length 𝑁 as words with particular endpoints (df-wspthsnon 27630): (𝐴(𝑁 WSPathsNOn 𝐺)𝐵)

* Closed Walks of a fixed length 𝑁 as words anchored at a particular vertex 𝐴 (df-wwlksnon 27628): (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 27398. 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 27399, restricting s to 1. wlk1ewlk 27439 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".

Syntaxcewlks 27395 Extend class notation with s-walks "on the edge level" (of a hypergraph).
class EdgWalks

Syntaxcwlks 27396 Extend class notation with walks (i.e. 1-walks) (of a hypergraph).
class Walks

Syntaxcwlkson 27397 Extend class notation with walks between two vertices (within a graph).
class WalksOn

Definitiondf-ewlks 27398* 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 27399* 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 27438) 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 27410.

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 27400* 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 27457. 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)}))

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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 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