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

Theoremuvtxnbvtxm1 26301 A universal vertex has 𝑛 − 1 neighbors in a finite simple graph with 𝑛 vertices. A biconditional version of nbusgrvtxm1uvtx 26300 resp. uvtxanm1nbgr 26299. (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 26302* 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 26303* 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)))

16.2.9.4  Complete graphs

Theoremiscplgr 26304* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Theoremcplgruvtxb 26305 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 26306* 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 26307* 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 26308 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Theoremcusgrusgr 26309 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 26310 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Theoremiscusgrvtx 26311* 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 26312 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 26313* 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 26314* 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 26315 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 26316 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 26317 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 26318 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 26319 Lemma for cplgr1v 26320 and cusgr1v 26321. (Contributed by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ V)

Theoremcplgr1v 26320 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 26321 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 26322 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 26323 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 26324 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 26325 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 26326* 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 26327 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

Theoremcusgrop 26328 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)

Theoremcusgrexilem1 26329* Lemma 1 for cusgrexi 26333. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ( I ↾ 𝑃) ∈ V)

Theoremusgrexilem 26330* Lemma for usgrexi 26331. (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 26331* 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 26332* Lemma 2 for cusgrexi 26333. (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 26333* 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.) (Proof shortened by AV, 10-Nov-2021.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}       (𝑉𝑊 → ⟨𝑉, ( I ↾ 𝑃)⟩ ∈ ComplUSGraph)

Theoremcusgrexg 26334* 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 26335* 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 26336* 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.)
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 26337* 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 26338* 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 26339 Base case of the induction in cusgrsize 26344. 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 26340 Base case of the induction in cusgrsize 26344. 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 26341* Lemma for cusgrsizeinds 26342. (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 26342* Part 1 of induction step in cusgrsize 26344. 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 26343* Induction step in cusgrsize 26344. 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 26344 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 26345* Lemma 1 for cusgrfi 26348. (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 26346* Lemma 2 for cusgrfi 26348. (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 26347* Lemma 3 for cusgrfi 26348. (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 26348 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 26349 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 26350* 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 26351 Lemma 1 for sizusglecusg 26353. (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 26352 Lemma 2 for sizusglecusg 26353. (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 26353 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 26354 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))

16.2.10  Vertex degree

Syntaxcvtxdg 26355 Extend class notation with the vertex degree function.
class VtxDeg

Definitiondf-vtxdg 26356* 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 26357* 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 26358* 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 26359* 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}})))

Theoremvtxdgop 26360 The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
(𝐺𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Theoremvtxdgf 26361 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 26362 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 26363 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 26364 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, for example in a Königsberg graph (see also the induction steps vdegp1ai 26426, vdegp1bi 26427 and vdegp1ci 26428). (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 26365 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 26366 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 26367 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 26368 The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 26364. (Contributed by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Theoremvtxdlfuhgr1v 26369* 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 26370 A vertex in a multigraph has degree 0 if the graph consists of only one vertex. (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)

Theoremvtxdun 26371 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 19-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → Fun 𝐽)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))       (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))

Theoremvtxdfiun 26372 The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → Fun 𝐽)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))    &   (𝜑 → dom 𝐼 ∈ Fin)    &   (𝜑 → dom 𝐽 ∈ Fin)       (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))

Theoremvtxduhgrun 26373 The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   (𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))       (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) +𝑒 ((VtxDeg‘𝐻)‘𝑁)))

Theoremvtxduhgrfiun 26374 The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐽 = (iEdg‘𝐻)    &   𝑉 = (Vtx‘𝐺)    &   (𝜑 → (Vtx‘𝐻) = 𝑉)    &   (𝜑 → (Vtx‘𝑈) = 𝑉)    &   (𝜑 → (dom 𝐼 ∩ dom 𝐽) = ∅)    &   (𝜑𝐺 ∈ UHGraph )    &   (𝜑𝐻 ∈ UHGraph )    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝑈) = (𝐼𝐽))    &   (𝜑 → dom 𝐼 ∈ Fin)    &   (𝜑 → dom 𝐽 ∈ Fin)       (𝜑 → ((VtxDeg‘𝑈)‘𝑁) = (((VtxDeg‘𝐺)‘𝑁) + ((VtxDeg‘𝐻)‘𝑁)))

Theoremvtxdlfgrval 26375* The value of the vertex degree function for a loop-free graph 𝐺. (Contributed by AV, 23-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐷 = (VtxDeg‘𝐺)       ((𝐼:𝐴⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)} ∧ 𝑈𝑉) → (𝐷𝑈) = (#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))

Theoremvtxdumgrval 26376* The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝑈𝑉) → (𝐷𝑈) = (#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))

Theoremvtxdusgrval 26377* The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐴 = dom 𝐼    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (𝐷𝑈) = (#‘{𝑥𝐴𝑈 ∈ (𝐼𝑥)}))

Theoremvtxd0nedgb 26378* A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝑈𝑉 → ((𝐷𝑈) = 0 ↔ ¬ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))

Theoremvtxdushgrfvedglem 26379* Lemma for vtxdushgrfvedg 26380 and vtxdusgrfvedg 26381. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USHGraph ∧ 𝑈𝑉) → (#‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (#‘{𝑒𝐸𝑈𝑒}))

Theoremvtxdushgrfvedg 26380* The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USHGraph ∧ 𝑈𝑉) → (𝐷𝑈) = ((#‘{𝑒𝐸𝑈𝑒}) +𝑒 (#‘{𝑒𝐸𝑒 = {𝑈}})))

Theoremvtxdusgrfvedg 26381* 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 26382* 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 26383* 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 26384* 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 26385* 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 26386* Alternate proof of vtxdusgr0edgnel 26385, not based on vtxduhgr0edgnel 26384. 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 26387 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 26388* 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 26389* 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 26390 A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 26135). (Contributed by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑𝐺 ∈ USPGraph )

Theorem1loopgredg 26391 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 26392 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 26393 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 26394 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 26395 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 26396 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 26397 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 26398 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 26399 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 26400 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)

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