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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnbgrnself2 26301 A class 𝑋 is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)
𝑋 ∉ (𝐺 NeighbVtx 𝑋)

Theoremnbgrssovtx 26302 The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself. Stronger version of nbgrssvtx 26281. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑋})

Theoremnbgrssvwo2 26303 The neighbors of a vertex 𝑋 form a subset of all vertices except the vertex 𝑋 itself and a class 𝑀 which is not a neighbor of 𝑋. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 12-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑀 ∉ (𝐺 NeighbVtx 𝑋) → (𝐺 NeighbVtx 𝑋) ⊆ (𝑉 ∖ {𝑀, 𝑋}))

Theoremnbgrnself2OLD 26304 Obsolete version of nbgrnself2 26301 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))

TheoremnbgrssovtxOLD 26305 Obsolete version of nbgrssovtx 26302 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑁}))

Theoremnbgrssvwo2OLD 26306 Obsolete version of nbgrssvwo2 26303 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝑀 ∉ (𝐺 NeighbVtx 𝑁)) → (𝐺 NeighbVtx 𝑁) ⊆ (𝑉 ∖ {𝑀, 𝑁}))

Theoremusgrnbnself2OLD 26307 Obsolete as of 12-Feb-2022. Use nbgrnself2 26301 instead. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ USGraph → 𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Theoremnbgrsym 26308 In a graph, the neighborhood relation is symmetric: a vertex 𝑁 in a graph 𝐺 is a neighbor of a second vertex 𝐾 iff the second vertex 𝐾 is a neighbor of the first vertex 𝑁. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Revised by AV, 12-Feb-2022.)
(𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁))

TheoremnbgrsymOLD 26309 Obsolete version of nbgrsym 26308 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ 𝐾 ∈ (𝐺 NeighbVtx 𝑁)))

Theoremnbupgrres 26310* The neighborhood of a vertex in a restricted pseudograph (not necessarily valid for a hypergraph, because 𝑁, 𝐾 and 𝑀 could be connected by one edge, so 𝑀 is a neighbor of 𝐾 in the original graph, but not in the restricted graph, because the edge between 𝑀 and 𝐾, also incident with 𝑁, was removed). (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}) ∧ 𝑀 ∈ (𝑉 ∖ {𝑁, 𝐾})) → (𝑀 ∈ (𝐺 NeighbVtx 𝐾) → 𝑀 ∈ (𝑆 NeighbVtx 𝐾)))

Theoremusgrnbcnvfv 26311 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → (𝐼‘(𝐼‘{𝐾, 𝑁})) = {𝐾, 𝑁})

Theoremnbusgredgeu 26312* 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.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑀 ∈ (𝐺 NeighbVtx 𝑁)) → ∃!𝑒𝐸 𝑒 = {𝑀, 𝑁})

Theoremedgnbusgreu 26313* For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑀)       (((𝐺 ∈ USGraph ∧ 𝑀𝑉) ∧ (𝐶𝐸𝑀𝐶)) → ∃!𝑛𝑁 𝐶 = {𝑀, 𝑛})

Theoremnbusgredgeu0 26314* 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.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑁 = (𝐺 NeighbVtx 𝑈)    &   𝐼 = {𝑒𝐸𝑈𝑒}       (((𝐺 ∈ USGraph ∧ 𝑈𝑉) ∧ 𝑀𝑁) → ∃!𝑖𝐼 𝑖 = {𝑈, 𝑀})

Theoremnbusgrf1o0 26315* 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 26316* 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 26317* 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 26318* 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 26319* 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 26320 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 26321 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 26322 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 26323 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.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 1))

Theoremnbfusgrlevtxm2 26324 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.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) ∧ (𝑀𝑉𝑀𝑈𝑀 ∉ (𝐺 NeighbVtx 𝑈))) → (#‘(𝐺 NeighbVtx 𝑈)) ≤ ((#‘𝑉) − 2))

Theoremnbusgrvtxm1 26325 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 26326 Lemma 1 for nb3grpr 26328. (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 26327* Lemma 2 for nb3grpr 26328. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.) (Proof shortened by AV, 13-Feb-2022.)
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 26328* 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 26329 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 26330 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})))

16.2.9.2  Universal vertices

Syntaxcuvtx 26331 Extend class notation with the universal vertices (in a graph).
class UnivVtx

Definitiondf-uvtx 26332* Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph), or equivalently, if all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.)
UnivVtx = (𝑔 ∈ V ↦ {𝑣 ∈ (Vtx‘𝑔) ∣ ∀𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣})𝑛 ∈ (𝑔 NeighbVtx 𝑣)})

Theoremuvtxval 26333* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (Revised by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}

TheoremuvtxavalOLD 26334* Obsolete version of uvtxel 26335 as of 14-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})

Theoremuvtxel 26335* 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.) (Revised by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁)))

TheoremuvtxaelOLD 26336* Obsolete version of uvtxel 26335 as of 14-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))))

Theoremuvtxisvtx 26337 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → 𝑁𝑉)

Theoremuvtxssvtx 26338 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 26339* A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) → ∀𝑛 ∈ (𝑉 ∖ {𝑁})𝑛 ∈ (𝐺 NeighbVtx 𝑁))

Theoremuvtxnbgrss 26340 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))

Theoremuvtxnbgrvtx 26341* 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))

Theoremuvtx0 26342 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.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)

Theoremisuvtx 26343* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Revised by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒𝐸 {𝑘, 𝑣} ⊆ 𝑒}

TheoremisuvtxaOLD 26344* Obsolete version of uvtxel1 26345 as of 14-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑣})∃𝑒𝐸 {𝑘, 𝑣} ⊆ 𝑒})

Theoremuvtxel1 26345* Characterization of a universal vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒))

Theoremuvtx01vtx 26346 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.) (Revised by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐸 = ∅ → ((UnivVtx‘𝐺) ≠ ∅ ↔ (#‘𝑉) = 1))

Theoremuvtxael1OLD 26347* Obsolete version of uvtxel1 26345 as of 14-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺𝑊 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁})∃𝑒𝐸 {𝑘, 𝑁} ⊆ 𝑒)))

Theoremuvtxa01vtx0OLD 26348 Obsolete version of uvtx01vtx 26346 as of 14-Feb-2022. (Contributed by AV, 30-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊𝐸 = ∅) → ((UnivVtx‘𝐺) ≠ ∅ ↔ (#‘𝑉) = 1))

Theoremuvtx2vtx1edg 26349* 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.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((#‘𝑉) = 2 ∧ 𝑉𝐸) → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Theoremuvtx2vtx1edgb 26350* 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 26351 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 26352 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.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝑁𝑉 → (𝑁 ∈ (UnivVtx‘𝐺) ↔ (𝐺 NeighbVtx 𝑁) = (𝑉 ∖ {𝑁})))

Theoremuvtxusgr 26353* 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 26354* 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)))

Theoremuvtxnm1nbgr 26355 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 26356 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.) (Proof shortened by AV, 13-Feb-2022.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → ((#‘(𝐺 NeighbVtx 𝑈)) = ((#‘𝑉) − 1) → 𝑈 ∈ (UnivVtx‘𝐺)))

Theoremuvtxnbvtxm1 26357 A universal vertex has 𝑛 − 1 neighbors in a finite simple graph with 𝑛 vertices. A biconditional version of nbusgrvtxm1uvtx 26356 resp. uvtxnm1nbgr 26355. (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 26358* 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.) (Proof shortened by AV, 13-Feb-2022.)
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 26359* 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)))

16.2.9.3  Complete graphs

Syntaxccplgr 26360 Extend class notation with (arbitrary) complete graphs.
class ComplGraph

Syntaxccusgr 26361 Extend class notation with complete simple graphs.
class ComplUSGraph

Definitiondf-cplgr 26362 Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e., each vertex has all other vertices as neighbors or, in other words, each vertex is a universal vertex. (Contributed by AV, 24-Oct-2020.) (Revised by TA, 15-Feb-2022.)
ComplGraph = {𝑔 ∣ (UnivVtx‘𝑔) = (Vtx‘𝑔)}

Definitiondf-cusgr 26363 Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 26406. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by BJ, 14-Feb-2022.)
ComplUSGraph = (USGraph ∩ ComplGraph)

Theoremcplgruvtxb 26364 A graph 𝐺 is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 15-Feb-2022.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Theoremprcliscplgr 26365* A proper class (representing a null graph, see vtxvalprc 25982) has the property of a complete graph (see also cplgr0v 26379), but cannot be an element of ComplGraph, of course. Because of this, a sethood antecedent like 𝐺𝑊 is necessary in the following theorems like iscplgr 26366. (Contributed by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       𝐺 ∈ V → ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺))

Theoremiscplgr 26366* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

TheoremcplgruvtxbOLD 26367 Obsolete proof of cplgruvtxb 26364 as of 15-Feb-2022. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ (UnivVtx‘𝐺) = 𝑉))

Theoremiscplgrnb 26368* 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 26369* A graph 𝐺 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 26370 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Theoremcusgrusgr 26371 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 26372 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Theoremiscusgrvtx 26373* 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 26374 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 26375* 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 26376* 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 26377 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 26378 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 26379 A null graph (with no vertices) 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 26380 A graph with no vertices and 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 26381 Lemma for cplgr1v 26382 and cusgr1v 26383. (Contributed by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ V)

Theoremcplgr1v 26382 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.) (Proof shortened by AV, 14-Feb-2022.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ ComplGraph)

Theoremcusgr1v 26383 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 26384 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 26385 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 26386 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 26387 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.) (Proof shortened by AV, 13-Feb-2022.)
𝐸 = (Edg‘𝐺)    &   (Vtx‘𝐺) = {𝐴, 𝐵, 𝐶}       (((𝐴𝑋𝐵𝑌𝐶𝑍) ∧ 𝐺 ∈ UPGraph ∧ (𝐴𝐵𝐴𝐶𝐵𝐶)) → (𝐺 ∈ ComplGraph ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸 ∧ {𝐶, 𝐴} ∈ 𝐸)))

Theoremcusgr3vnbpr 26388* 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 26389 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

Theoremcusgrop 26390 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)

Theoremcusgrexilem1 26391* Lemma 1 for cusgrexi 26395. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)

Theoremusgrexilem 26392* Lemma for usgrexi 26393. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ( I ↾ 𝑃):dom ( I ↾ 𝑃)–1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremusgrexi 26393* 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.) (Proof shortened by AV, 10-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ USGraph)

Theoremcusgrexilem2 26394* Lemma 2 for cusgrexi 26395. (Contributed by AV, 12-Jan-2018.) (Revised by AV, 10-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (((𝑉𝑊𝑣𝑉) ∧ 𝑛 ∈ (𝑉 ∖ {𝑣})) → ∃𝑒 ∈ ran ( I ↾ 𝑃){𝑣, 𝑛} ⊆ 𝑒)

Theoremcusgrexi 26395* An arbitrary set 𝑉 regarded as set of 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.) (Proof shortened by AV, 14-Feb-2022.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Theoremcusgrexg 26396* For each set there is a set of edges so that the set together with these edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.)
(𝑉𝑊 → ∃𝑒𝑉, 𝑒⟩ ∈ ComplUSGraph)

Theoremstructtousgr 26397* Any (extensible) structure with a base set can be made a simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (#‘𝑥) = 2}    &   (𝜑𝑆 Struct 𝑋)    &   𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑆)       (𝜑𝐺 ∈ USGraph)

Theoremstructtocusgr 26398* Any (extensible) structure with a base set can be made a complete simple graph with the set of pairs of elements of the base set regarded as edges. (Contributed by AV, 10-Nov-2021.) (Revised by AV, 17-Nov-2021.) (Proof shortened by AV, 14-Feb-2022.)
𝑃 = {𝑥 ∈ 𝒫 (Base‘𝑆) ∣ (#‘𝑥) = 2}    &   (𝜑𝑆 Struct 𝑋)    &   𝐺 = (𝑆 sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)    &   (𝜑 → (Base‘ndx) ∈ dom 𝑆)       (𝜑𝐺 ∈ ComplUSGraph)

Theoremcffldtocusgr 26399* The field of complex numbers can be made a complete simple graph with the set of pairs of complex numbers regarded as edges. This theorem demonstrates the capabilities of the current definitions for graphs applied to extensible structures. (Contributed by AV, 14-Nov-2021.) (Proof shortened by AV, 17-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 ℂ ∣ (#‘𝑥) = 2}    &   𝐺 = (ℂfld sSet ⟨(.ef‘ndx), ( I ↾ 𝑃)⟩)       𝐺 ∈ ComplUSGraph

Theoremcusgrres 26400* 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)

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