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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvtxdusgrfvedg 27201* The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (𝐷𝑈) = (♯‘{𝑒𝐸𝑈𝑒}))
 
Theoremvtxduhgr0nedg 27202* If a vertex in a hypergraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉 ∧ (𝐷𝑈) = 0) → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ 𝐸)
 
Theoremvtxdumgr0nedg 27203* If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑈𝑉 ∧ (𝐷𝑈) = 0) → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ 𝐸)
 
Theoremvtxduhgr0edgnel 27204* A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → ((𝐷𝑈) = 0 ↔ ¬ ∃𝑒𝐸 𝑈𝑒))
 
Theoremvtxdusgr0edgnel 27205* A vertex in a simple graph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐷𝑈) = 0 ↔ ¬ ∃𝑒𝐸 𝑈𝑒))
 
Theoremvtxdusgr0edgnelALT 27206* Alternate proof of vtxdusgr0edgnel 27205, not based on vtxduhgr0edgnel 27204. A vertex in a simple graph has degree 0 if there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐷𝑈) = 0 ↔ ¬ ∃𝑒𝐸 𝑈𝑒))
 
Theoremvtxdgfusgrf 27207 The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → (VtxDeg‘𝐺):𝑉⟶ℕ0)
 
Theoremvtxdgfusgr 27208* In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 12-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) ∈ ℕ0)
 
Theoremfusgrn0degnn0 27209* In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 1-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → ∃𝑣𝑉𝑛 ∈ ℕ0 ((VtxDeg‘𝐺)‘𝑣) = 𝑛)
 
Theorem1loopgruspgr 27210 A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 26959). (Contributed by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑𝐺 ∈ USPGraph)
 
Theorem1loopgredg 27211 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 27212 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 27213 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 27214 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 27215 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 27216 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 27217 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 27218 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 27219 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 27220 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 27221 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 27222 Lemma for p1evtxdeq 27223 and p1evtxdp1 27224. (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 27223 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 27224 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 27225 The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26959). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)
 
Theoremuspgrloopvtxel 27226 A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26959). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))
 
Theoremuspgrloopiedg 27227 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26959) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})
 
Theoremuspgrloopedg 27228 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26959) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
 
Theoremuspgrloopnb0 27229 In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26959), 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 27230 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 26959), 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 27231 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 27232 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 27233 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 27234 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 27235 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 27236 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 27237 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 27238 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 27229, but degree 2, see uspgrloopvd2 27230. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 27236, but also degree 2, see umgr2v2evd2 27237. (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 27239 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 27240* 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 27241* 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 27242* 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 27243* 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 27244* 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 27245* 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 27246* 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 27247* 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 27248* 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 27249 Lemma 1 for vtxdginducedm1 27253: 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 27250* Lemma 2 for vtxdginducedm1 27253: 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 27251* Lemma 3 for vtxdginducedm1 27253: 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 27252* Lemma 4 for vtxdginducedm1 27253. (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 27253* 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 27254* 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 27255* Lemma for finsumvtxdg2sstep 27259. (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 27256* Lemma for finsumvtxdg2sstep 27259. (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 27257* Lemma for finsumvtxdg2sstep 27259. (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 27258* Lemma for finsumvtxdg2sstep 27259. (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 27259* Induction step of finsumvtxdg2size 27260: 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 27260* 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 27261) 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 27261* 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 27262* 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 27263* 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 27264* 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 27267 and df-rusgr 27268, 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 27304), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 27302). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).

 
Syntaxcrgr 27265 Extend class notation to include the class of all regular graphs.
class RegGraph
 
Syntaxcrusgr 27266 Extend class notation to include the class of all regular simple graphs.
class RegUSGraph
 
Definitiondf-rgr 27267* 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 27268* 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 27269* 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 27270* 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 27271 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 27272 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 27273 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 27274 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 27275 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 27276* 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 27277* 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 27278* 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 27279* 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 27280* 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 27281 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 27282 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 27283 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0)
 
Theoremuhgr0edg0rgrb 27284 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 27285 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 27286* 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 27287* 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 27288* 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 27289 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 27290 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 27291 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 27292 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 27293* 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 27294* 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 27295* 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 27296* 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 27297* Lemma for rusgr1vtx 27298. (Contributed by AV, 27-Dec-2020.)
(((∀𝑣𝑉 (♯‘𝐴) = 𝐾 ∧ ∀𝑣𝑉 𝐴 = ∅) ∧ (𝑉𝑊 ∧ (♯‘𝑉) = 1)) → 𝐾 = 0)
 
Theoremrusgr1vtx 27298 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 27299* 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 27300 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegUSGraph 0} ∉ V
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