Home Metamath Proof ExplorerTheorem List (p. 407 of 424) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27159) Hilbert Space Explorer (27160-28684) Users' Mathboxes (28685-42360)

Theorem List for Metamath Proof Explorer - 40601-40700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremedgusgrnbfin 40601* 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 40602 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 40603 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 40604 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 40605 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 40606 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 40607 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 40608 Lemma 1 for nb3grapr 25982. (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 40609* Lemma 2 for nb3grapr 25982. (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 40610* 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 40611 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 40612 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 40613* 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 40614* 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 40615 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 40616 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 40617* 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 40618 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 40619* 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 40620 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 40621* 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 40622* 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 40623 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 40624 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 40625* 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 40626* 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 40627 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 40628 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 40629* 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 40630* 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 40631 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 40632 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 40633 A universal vertex has 𝑛 − 1 neighbors in a finite simple graph with 𝑛 vertices. A biconditional version of nbusgrvtxm1uvtx 40632 resp. uvtxanm1nbgr 40631. (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 40634* 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 40635* 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 40636* The property of being a complete graph. (Contributed by AV, 1-Nov-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺𝑊 → (𝐺 ∈ ComplGraph ↔ ∀𝑣𝑉 𝑣 ∈ (UnivVtx‘𝐺)))

Theoremcplgruvtxb 40637 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 40638* 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 40639* 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 40640 The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝐺 ∈ ComplGraph))

Theoremcusgrusgr 40641 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 40642 A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.)
(𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph)

Theoremiscusgrvtx 40643* 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 40644 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 40645* 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 40646* 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 40647 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 40648 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 40649 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 40650 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 40651 Lemma for cplgr1v 40652 and cusgr1v 40653. (Contributed by AV, 23-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((#‘𝑉) = 1 → 𝐺 ∈ V)

Theoremcplgr1v 40652 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 40653 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 40654 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 40655 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 40656 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 40657 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 40658* 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 40659 A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplGraph)

Theoremcusgrop 40660 A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.)
(𝐺 ∈ ComplUSGraph → ⟨(Vtx‘𝐺), (iEdg‘𝐺)⟩ ∈ ComplUSGraph)

Theoremusgrexi 40661* 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 40662* 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 40663* 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 40664* 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 40665 Base case of the induction in cusgrasize 26006. 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 40666 Base case of the induction in cusgrasize 26006. 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 40667* Lemma for cusgrsizeinds 40668. (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 40668* Part 1 of induction step in cusgrsize 40670. 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 40669* Induction step in cusgrasize 26006. 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 40670 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 40671* Lemma 1 for cusgrfi 40674. (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 40672* Lemma 2 for cusgrfi 40674. (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 40673* Lemma 3 for cusgrfi 40674. (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 40674 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 40675 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 40676* 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 40677 Lemma 1 for sizusglecusg 40679. (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 40678 Lemma 2 for sizusglecusg 40679. (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 40679 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 40680 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))

21.34.8.7  Vertex degree

The definition df-vdgr 26421 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 40681 Extend class notation with the vertex degree function.
class VtxDeg

Definitiondf-vtxdg 40682* 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 40683* 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 40684* 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 40685* 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 40686 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 40687 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 40688 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 40689 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 40752, vdegp1ai-av 40752 and vdegp1ai-av 40752). (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 40690 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 40691 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 40692 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 40693 The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 40689. (Contributed by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Theoremvtxdlfuhgr1v 40694* 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 40695 A vertex in a multigraph has degree 0 if the graph consists of only one vertex. Formerly vdfrgra0 26549. (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 40696 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 40697 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 40698 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 40699 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 40700* 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)}))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
 Copyright terms: Public domain < Previous  Next >