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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnbusgredgeu0 40701* For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑈)    &   𝐼 = {𝑒𝐸𝑈𝑒}       (((𝐺 ∈ USGraph ∧ 𝑈𝑉) ∧ 𝑀𝑁) → ∃!𝑖𝐼 𝑖 = {𝑈, 𝑀})
 
Theoremnbusgrf1o0 40702* The mapping of neighbors of a vertex to edges incident to the vertex is a bijection ( 1-1 onto function) in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑈)    &   𝐼 = {𝑒𝐸𝑈𝑒}    &   𝐹 = (𝑛𝑁 ↦ {𝑈, 𝑛})       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → 𝐹:𝑁1-1-onto𝐼)
 
Theoremnbusgrf1o1 40703* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑈)    &   𝐼 = {𝑒𝐸𝑈𝑒}       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ∃𝑓 𝑓:𝑁1-1-onto𝐼)
 
Theoremnbusgrf1o 40704* The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ∃𝑓 𝑓:(𝐺 NeighbVtx 𝑈)–1-1-onto→{𝑒𝐸𝑈𝑒})
 
Theoremnbedgusgr 40705* The number of neighbors of a vertex is the number of edges at the vertex in a simple graph. (Contributed by AV, 27-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (#‘(𝐺 NeighbVtx 𝑈)) = (#‘{𝑒𝐸𝑈𝑒}))
 
Theoremedgusgrnbfin 40706* The number of neighbors of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((𝐺 NeighbVtx 𝑈) ∈ Fin ↔ {𝑒𝐸𝑈𝑒} ∈ Fin))
 
Theoremnbusgrfi 40707 The class of neighbors of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐸 ∈ Fin ∧ 𝑈𝑉) → (𝐺 NeighbVtx 𝑈) ∈ Fin)
 
Theoremnbfiusgrfi 40708 The class of neighbors of a vertex in a finite simple graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.) (Revised by AV, 28-Oct-2020.)
((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (Vtx‘𝐺)) → (𝐺 NeighbVtx 𝑁) ∈ Fin)
 
Theoremhashnbusgrnn0 40709 The number of neighbors of a vertex in a finite simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (#‘(𝐺 NeighbVtx 𝑈)) ∈ ℕ0)
 
Theoremnbfusgrlevtxm1 40710 The number of neighbors of a vertex is at most the number of vertices of the graph minus 1 in a finite simple graph. (Contributed by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 1))
 
Theoremnbfusgrlevtxm2 40711 If there is a vertex which is not a neighbor of another vertex, the number of neighbors of the other vertex is at most the number of vertices of the graph minus 2 in a finite simple graph. (Contributed by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))
 
Theoremnbusgrvtxm1 40712 If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → ((𝑀𝑉𝑀𝑈) → 𝑀 ∈ (𝐺 NeighbVtx 𝑈))))
 
Theoremnb3grprlem1 40713 Lemma 1 for nb3grapr 25720. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑𝐺 ∈ USGraph )    &   (𝜑𝑉 = {𝐴, 𝐵, 𝐶})    &   (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))       (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐴, 𝐶} ∈ 𝐸)))
 
Theoremnb3grprlem2 40714* Lemma 2 for nb3grapr 25720. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑𝐺 ∈ USGraph )    &   (𝜑𝑉 = {𝐴, 𝐵, 𝐶})    &   (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))    &   (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))       (𝜑 → ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ↔ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣})(𝐺 NeighbVtx 𝐴) = {𝑣, 𝑤}))
 
Theoremnb3grpr 40715* The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑𝐺 ∈ USGraph )    &   (𝜑𝑉 = {𝐴, 𝐵, 𝐶})    &   (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))    &   (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))       (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
 
Theoremnb3grpr2 40716 The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   (𝜑𝐺 ∈ USGraph )    &   (𝜑𝑉 = {𝐴, 𝐵, 𝐶})    &   (𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))    &   (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))       (𝜑 → (({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
 
Theoremnb3gr2nb 40717 If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.)
(((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ ((Vtx‘𝐺) = {𝐴, 𝐵, 𝐶} ∧ 𝐺 ∈ USGraph )) → (((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶}) ↔ ((𝐺 NeighbVtx 𝐴) = {𝐵, 𝐶} ∧ (𝐺 NeighbVtx 𝐵) = {𝐴, 𝐶} ∧ (𝐺 NeighbVtx 𝐶) = {𝐴, 𝐵})))
 
Theoremuvtxaval 40718* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
 
Theoremuvtxael 40719* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))))
 
Theoremuvtxaisvtx 40720 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁𝑉)
 
Theoremuvtxassvtx 40721 The set of the universal vertices is a subset of the set of all vertices. (Contributed by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (UnivVtx‘𝐺) ⊆ 𝑉
 
Theoremvtxnbuvtx 40722* A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))
 
Theoremuvtxanbgr 40723 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → (𝑉 ∖ {𝑁}) ⊆ (𝐺 NeighbVtx 𝑁))
 
Theoremuvtxanbgrvtx 40724* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (𝐺 NeighbVtx 𝑣))
 
Theoremuvtxa0 40725 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)       (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)
 
Theoremisuvtxa 40726* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒𝐸 {𝑘, 𝑣} ⊆ 𝑒})
 
Theoremuvtxael1 40727* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒)))
 
Theoremuvtxa01vtx0 40728 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊𝐸 = ∅) → ((UnivVtx‘𝐺) ≠ ∅ ↔ (#‘𝑉) = 1))
 
Theoremuvtxa01vtx 40729 If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (#‘𝑉) = 1))
 
Theoremuvtx2vtx1edg 40730* If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 25-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((#‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))
 
Theoremuvtx2vtx1edgb 40731* If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 2) → (𝑉𝐸 ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
 
Theoremuvtxnbgr 40732 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
 
Theoremuvtxnbgrb 40733 A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))
 
Theoremuvtxusgr 40734* The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (UnivVtx‘𝐺) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ 𝐸})
 
Theoremuvtxusgrel 40735* A universal vertex, i.e. an element of the set of all universal vertices, of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ 𝐸)))
 
Theoremuvtxanm1nbgr 40736 A universal vertex has 𝑛 − 1 neighbors in a finite graph with 𝑛 vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ (UnivVtx‘𝐺)) → (#‘(𝐺 NeighbVtx 𝑁)) = ((#‘𝑉) − 1))
 
Theoremnbusgrvtxm1uvtx 40737 If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → 𝑈 ∈ (UnivVtx‘𝐺)))
 
Theoremuvtxnbvtxm1 40738 A universal vertex has 𝑛 − 1 neighbors in a finite simple graph with 𝑛 vertices. A biconditional version of nbusgrvtxm1uvtx 40737 resp. uvtxanm1nbgr 40736. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (𝑈 ∈ (UnivVtx‘𝐺) ↔ (#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1)))
 
Theoremnbupgruvtxres 40739* The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → ((𝐺 NeighbVtx 𝐾) = (𝑉 ∖ {𝐾}) → (𝑆 NeighbVtx 𝐾) = (𝑉 ∖ {𝑁, 𝐾})))
 
Theoremuvtxupgrres 40740* A universal vertex is universal in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ 𝐾 ∈ (𝑉 ∖ {𝑁})) → (𝐾 ∈ (UnivVtx‘𝐺) → 𝐾 ∈ (UnivVtx‘𝑆)))
 
Theoremiscplgr 40741* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
 
Theoremcplgruvtxb 40742 An graph is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))
 
Theoremiscplgrnb 40743* A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)))
 
Theoremiscplgredg 40744* A graph is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉𝑛 ∈ (𝑉 ∖ {𝑣})∃𝑒𝐸 {𝑣, 𝑛} ⊆ 𝑒))
 
Theoremiscusgr 40745 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))
 
Theoremcusgrusgr 40746 A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )
 
Theoremcusgrcplgr 40747 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)
 
Theoremiscusgrvtx 40748* A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))
 
Theoremcusgruvtxb 40749 A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ USGraph → (𝐺 ∈ ComplUSGraph ↔ (UnivVtx‘𝐺) = 𝑉))
 
Theoremiscusgredg 40750* A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ ∀𝑘𝑉𝑛 ∈ (𝑉 ∖ {𝑘}){𝑛, 𝑘} ∈ 𝐸))
 
Theoremcusgredg 40751* In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ ComplUSGraph → 𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
 
Theoremcplgr0 40752 The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.)
∅ ∈ ComplGraph
 
Theoremcusgr0 40753 The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.)
∅ ∈ ComplUSGraph
 
Theoremcplgr0v 40754 A graph with no vertices (and therefore no edges) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑉 = ∅) → 𝐺 ∈ ComplGraph)
 
Theoremcusgr0v 40755 A graph with no vertices (and therefore no edges) is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑉 = ∅ ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph)
 
Theoremcplgr1vlem 40756 Lemma for cplgr1v 40757 and cusgr1v 40758. (Contributed by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ V)
 
Theoremcplgr1v 40757 A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ ComplGraph)
 
Theoremcusgr1v 40758 A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.) (Revised by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((#‘𝑉) = 1 ∧ (iEdg‘𝐺) = ∅) → 𝐺 ∈ ComplUSGraph)
 
Theoremcplgr2v 40759 An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 2) → (𝐺 ∈ ComplGraph ↔ 𝑉𝐸))
 
Theoremcplgr2vpr 40760 An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐴𝑋𝐵𝑌𝐴𝐵) ∧ (𝐺 ∈ UHGraph ∧ 𝑉 = {𝐴, 𝐵})) → (𝐺 ∈ ComplGraph ↔ {𝐴, 𝐵} ∈ 𝐸))
 
Theoremnbcplgr 40761 In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ ComplGraph ∧ 𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁}))
 
Theoremcplgr3v 40762 A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.)
𝐸 = (Edg‘𝐺)    &   (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))
 
Theoremcusgr3vnbpr 40763* The neighbors of a vertex in a simple graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 5-Nov-2020.)
𝐸 = (Edg‘𝐺)    &   (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}    &   𝑉 = (Vtx‘𝐺)       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ USGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ∀𝑥𝑉𝑦𝑉𝑧 ∈ (𝑉 ∖ {𝑦})(𝐺 NeighbVtx 𝑥) = {𝑦, 𝑧}))
 
Theoremcplgrop 40764 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)
 
Theoremcusgrop 40765 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)
 
Theoremusgrexi 40766* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph )
 
Theoremcusgrexi 40767* An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)
 
Theoremcusgrexg 40768* For each set there is a set of edges so that the set together with these edges is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
(𝑉𝑊 → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)
 
Theoremcusgrres 40769* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}    &   𝑆 = ⟨(𝑉 ∖ {𝑁}), ( I ↾ 𝐹)⟩       ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑆 ∈ ComplUSGraph)
 
Theoremcusgrsizeindb0 40770 Base case of the induction in cusgrasize 25744. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 0) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrsizeindb1 40771 Base case of the induction in cusgrasize 25744. The size of a (complete) simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ (#‘𝑉) = 1) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrsizeindslem 40772* Lemma for cusgrsizeinds 40773. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘{𝑒𝐸𝑁𝑒}) = ((#‘𝑉) − 1))
 
Theoremcusgrsizeinds 40773* Part 1 of induction step in cusgrsize 40775. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹)))
 
Theoremcusgrsize2inds 40774* Induction step in cusgrasize 25744. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐹 = {𝑒𝐸𝑁𝑒}       (𝑌 ∈ ℕ0 → ((𝐺 ∈ ComplUSGraph ∧ (#‘𝑉) = 𝑌𝑁𝑉) → ((#‘𝐹) = ((#‘(𝑉 ∖ {𝑁}))C2) → (#‘𝐸) = ((#‘𝑉)C2))))
 
Theoremcusgrsize 40775 The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2), see definition in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 10-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))
 
Theoremcusgrfilem1 40776* Lemma 1 for cusgrfi 40779. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}       ((𝐺 ∈ ComplUSGraph ∧ 𝑁𝑉) → 𝑃 ⊆ (Edg‘𝐺))
 
Theoremcusgrfilem2 40777* Lemma 2 for cusgrfi 40779. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       (𝑁𝑉𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)
 
Theoremcusgrfilem3 40778* Lemma 3 for cusgrfi 40779. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       (𝑁𝑉 → (𝑉 ∈ Fin ↔ 𝑃 ∈ Fin))
 
Theoremcusgrfi 40779 If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 𝐸 ∈ Fin) → 𝑉 ∈ Fin)
 
Theoremusgredgsscusgredg 40780 A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐻)    &   𝐹 = (Edg‘𝐻)       ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → 𝐸𝐹)
 
Theoremusgrsscusgr 40781* A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐻)    &   𝐹 = (Edg‘𝐻)       ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∀𝑒𝐸𝑓𝐹 𝑒 = 𝑓)
 
Theoremsizusglecusglem1 40782 Lemma 1 for sizusglecusg 40784. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐻)    &   𝐹 = (Edg‘𝐻)       ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸1-1𝐹)
 
Theoremsizusglecusglem2 40783 Lemma 2 for sizusglecusg 40784. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐻)    &   𝐹 = (Edg‘𝐻)       ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin)
 
Theoremsizusglecusg 40784 The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑉 = (Vtx‘𝐻)    &   𝐹 = (Edg‘𝐻)       ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (#‘𝐸) ≤ (#‘𝐹))
 
Theoremfusgrmaxsize 40785 The maximum size of a finite simple graph with 𝑛 vertices is (((𝑛 − 1)∗𝑛) / 2). See statement in section I.1 of [Bollobas] p. 3 . (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 14-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ FinUSGraph → (#‘𝐸) ≤ ((#‘𝑉)C2))
 
20.34.8.12  Vertex degree

The definition df-vdgr 26159 of the vertex degree VDeg is independent of the representation (or even the existence) of a graph. Therefore, it could be used for the revised definitions of graphs without modification. The way to use the set of vertices and the edge function separately differs from the way to use a class 𝐺 representing a graph and using the functions Vtx and iEdg, therefore, an alternate definition and related theorems are provided in this section.

 
Syntaxcvtxdg 40786 Extend class notation with the vertex degree function.
class VtxDeg
 
Definitiondf-vtxdg 40787* Define the vertex degree function for a graph. To be appropriate for arbitrary hypergraphs, we have to double-count those edges that contain 𝑢 "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition +𝑒 is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
VtxDeg = (𝑔 ∈ V ↦ (Vtx‘𝑔) / 𝑣(iEdg‘𝑔) / 𝑒(𝑢𝑣 ↦ ((#‘{𝑥 ∈ dom 𝑒𝑢 ∈ (𝑒𝑥)}) +𝑒 (#‘{𝑥 ∈ dom 𝑒 ∣ (𝑒𝑥) = {𝑢}}))))
 
Theoremvtxdgfval 40788* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼       (𝐺𝑊 → (VtxDeg‘𝐺) = (𝑢𝑉 ↦ ((#‘{𝑥𝐴𝑢 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑢}}))))
 
Theoremvtxdgval 40789* The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼       (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) +𝑒 (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
 
Theoremvtxdgfival 40790* The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼       ((𝐴 ∈ Fin ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = ((#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}) + (#‘{𝑥𝐴 ∣ (𝐼𝑥) = {𝑈}})))
 
Theoremvtxdgf 40791 The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (VtxDeg‘𝐺):𝑉⟶ℕ0*)
 
Theoremvtxdgelxnn0 40792 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (𝑋𝑉 → ((VtxDeg‘𝐺)‘𝑋) ∈ ℕ0*)
 
Theoremvtxdg0v 40793 The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 = ∅ ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = 0)
 
Theoremvtxdg0e 40794 The degree of a vertex in an empty graph is zero, because there are no edges. This is the base case for the induction for calculating the degree of a vertex (see also vdegp1ai-av 40857, vdegp1ai-av 40857 and vdegp1ai-av 40857). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝑈𝑉𝐼 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
 
Theoremvtxdgfisnn0 40795 The degree of a vertex in a graph of finite size is a nonnegative integer. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 11-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼       ((𝐴 ∈ Fin ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) ∈ ℕ0)
 
Theoremvtxdgfisf 40796 The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼       ((𝐺𝑊𝐴 ∈ Fin) → (VtxDeg‘𝐺):𝑉⟶ℕ0)
 
Theoremvtxdeqd 40797 Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
(𝜑𝐺𝑋)    &   (𝜑𝐻𝑌)    &   (𝜑 → (Vtx‘𝐻) = (Vtx‘𝐺))    &   (𝜑 → (iEdg‘𝐻) = (iEdg‘𝐺))       (𝜑 → (VtxDeg‘𝐻) = (VtxDeg‘𝐺))
 
Theoremvtxduhgr0e 40798 The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 40794. (Contributed by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)
 
Theoremvtxdlfuhgr1v 40799* The degree of the vertex in a loop-free hypergraph with one vertex is 0. (Contributed by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐸 = {𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)}       ((𝐺 ∈ UHGraph ∧ (#‘𝑉) = 1 ∧ 𝐼:dom 𝐼𝐸) → (𝑈𝑉 → ((VtxDeg‘𝐺)‘𝑈) = 0))
 
Theoremvdumgr0 40800 A vertex in a multigraph has degree 0 if the graph consists of only one vertex. Formerly vdfrgra0 26287. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 2-Apr-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑁𝑉 ∧ (#‘𝑉) = 1) → ((VtxDeg‘𝐺)‘𝑁) = 0)
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