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Theorem List for Metamath Proof Explorer - 26401-26500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcusgrsizeindb0 26401 Base case of the induction in cusgrsize 26406. 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 26402 Base case of the induction in cusgrsize 26406. 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 26403* Lemma for cusgrsizeinds 26404. (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 26404* Part 1 of induction step in cusgrsize 26406. 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 26405* Induction step in cusgrsize 26406. 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 26406 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 26407* Lemma 1 for cusgrfi 26410. (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 26408* Lemma 2 for cusgrfi 26410. (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 26409* Lemma 3 for cusgrfi 26410. (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 26410 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 26411 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 26412* 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 26413 Lemma 1 for sizusglecusg 26415. (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 26414 Lemma 2 for sizusglecusg 26415. (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 26415 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 26416 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 26417 Extend class notation with the vertex degree function.
class VtxDeg

Definitiondf-vtxdg 26418* 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 26419* 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 26420* 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 26421* 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 26422 The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)
(𝐺𝑊 → (VtxDeg‘𝐺) = ((Vtx‘𝐺)VtxDeg(iEdg‘𝐺)))

Theoremvtxdgf 26423 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 26424 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 26425 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 26426 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 26488, vdegp1bi 26489 and vdegp1ci 26490). (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 26427 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 26428 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 26429 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 26430 The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 26426. (Contributed by AV, 15-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉𝐸 = ∅) → ((VtxDeg‘𝐺)‘𝑈) = 0)

Theoremvtxdlfuhgr1v 26431* 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 26432 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 26433 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 26434 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 26435 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 26436 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 26437* 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 26438* 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 26439* 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 26440* 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 26441* Lemma for vtxdushgrfvedg 26442 and vtxdusgrfvedg 26443. (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 26442* 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 26443* 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 26444* 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 26445* 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 26446* 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 26447* 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 26448* Alternate proof of vtxdusgr0edgnel 26447, not based on vtxduhgr0edgnel 26446. 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 26449 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 26450* 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 26451* 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 26452 A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 26186). (Contributed by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑𝐺 ∈ USPGraph)

Theorem1loopgredg 26453 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 26454 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 26455 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 26456 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 26457 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 26458 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 26459 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 26460 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 26461 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 26462 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐶}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 1)

Theorem1egrvtxdg0 26463 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐷𝑉)    &   (𝜑𝐶𝐷)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝐵, 𝐷}⟩})       (𝜑 → ((VtxDeg‘𝐺)‘𝐶) = 0)

Theoremp1evtxdeqlem 26464 Lemma for p1evtxdeq 26465 and p1evtxdp1 26466. (Contributed by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸𝑌)       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 ((VtxDeg‘⟨𝑉, {⟨𝐾, 𝐸⟩}⟩)‘𝑈)))

Theoremp1evtxdeq 26465 If an edge 𝐸 which does not contain vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸𝑌)    &   (𝜑𝑈𝐸)       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = ((VtxDeg‘𝐺)‘𝑈))

Theoremp1evtxdp1 26466 If an edge 𝐸 (not being a loop) which contains vertex 𝑈 is added to a graph 𝐺 (yielding a graph 𝐹), the degree of 𝑈 is increased by 1. (Contributed by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑 → (Vtx‘𝐹) = 𝑉)    &   (𝜑 → (iEdg‘𝐹) = (𝐼 ∪ {⟨𝐾, 𝐸⟩}))    &   (𝜑𝐾𝑋)    &   (𝜑𝐾 ∉ dom 𝐼)    &   (𝜑𝑈𝑉)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑𝑈𝐸)    &   (𝜑 → 2 ≤ (#‘𝐸))       (𝜑 → ((VtxDeg‘𝐹)‘𝑈) = (((VtxDeg‘𝐺)‘𝑈) +𝑒 1))

Theoremuspgrloopvtx 26467 The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)

Theoremuspgrloopvtxel 26468 A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuspgrloopiedg 26469 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Theoremuspgrloopedg 26470 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Theoremuspgrloopnb0 26471 In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186), the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋𝑁𝑉) → (𝐺 NeighbVtx 𝑁) = ∅)

Theoremuspgrloopvd2 26472 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 26186), the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋𝑁𝑉) → ((VtxDeg‘𝐺)‘𝑁) = 2)

Theoremumgr2v2evtx 26473 The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)

Theoremumgr2v2evtxel 26474 A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       ((𝑉𝑊𝐴𝑉) → 𝐴 ∈ (Vtx‘𝐺))

Theoremumgr2v2eiedg 26475 The edge function in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       ((𝑉𝑊𝐴𝑉𝐵𝑉) → (iEdg‘𝐺) = {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩})

Theoremumgr2v2eedg 26476 The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       ((𝑉𝑊𝐴𝑉𝐵𝑉) → (Edg‘𝐺) = {{𝐴, 𝐵}})

Theoremumgr2v2e 26477 A multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → 𝐺 ∈ UMGraph)

Theoremumgr2v2enb1 26478 In a multigraph with two edges connecting the same two vertices, each of the vertices has one neighbor. (Contributed by AV, 18-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐺 NeighbVtx 𝐴) = {𝐵})

Theoremumgr2v2evd2 26479 In a multigraph with two edges connecting the same two vertices, each of the vertices has degree 2. (Contributed by AV, 18-Dec-2020.)
𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩       (((𝑉𝑊𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ((VtxDeg‘𝐺)‘𝐴) = 2)

Theoremhashnbusgrvd 26480 In a simple graph, the number of neighbors of a vertex is the degree of this vertex. This theorem does not hold for (simple) pseudographs, because a vertex connected with itself only by a loop has no neighbors, see uspgrloopnb0 26471, but degree 2, see uspgrloopvd2 26472. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 26478, but also degree 2, see umgr2v2evd2 26479. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (#‘(𝐺 NeighbVtx 𝑈)) = ((VtxDeg‘𝐺)‘𝑈))

Theoremusgruvtxvdb 26481 In a finite simple graph with n vertices a vertex is universal iff the vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑈𝑉) → (𝑈 ∈ (UnivVtx‘𝐺) ↔ ((VtxDeg‘𝐺)‘𝑈) = ((#‘𝑉) − 1)))

Theoremvdiscusgrb 26482* A finite simple graph with n vertices is complete iff every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → (𝐺 ∈ ComplUSGraph ↔ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1)))

Theoremvdiscusgr 26483* In a finite complete simple graph with n vertices every vertex has degree 𝑛 − 1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 ∈ FinUSGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = ((#‘𝑉) − 1) → 𝐺 ∈ ComplUSGraph))

Theoremvtxdusgradjvtx 26484* The degree of a vertex in a simple graph is the number of vertices adjacent to this vertex. (Contributed by Alexander van der Vekens, 9-Jul-2018.) (Revised by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → ((VtxDeg‘𝐺)‘𝑈) = (#‘{𝑣𝑉 ∣ {𝑈, 𝑣} ∈ 𝐸}))

Theoremusgrvd0nedg 26485* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 16-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝑈𝑉) → (((VtxDeg‘𝐺)‘𝑈) = 0 → ¬ ∃𝑣𝑉 {𝑈, 𝑣} ∈ 𝐸))

Theoremuhgrvd00 26486* If every vertex in a hypergraph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ UHGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))

Theoremusgrvd00 26487* If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝐺 ∈ USGraph → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 0 → 𝐸 = ∅))

Theoremvdegp1ai 26488* The induction step for a vertex degree calculation. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑌} to the edge set, where 𝑋𝑈𝑌, yields degree 𝑃 as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   𝑌𝑉    &   𝑌𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑌}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = 𝑃

Theoremvdegp1bi 26489* The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑈, 𝑋} to the edge set, where 𝑋𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑈, 𝑋}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)

Theoremvdegp1ci 26490* The induction step for a vertex degree calculation, for example in the Königsberg graph. If the degree of 𝑈 in the edge set 𝐸 is 𝑃, then adding {𝑋, 𝑈} to the edge set, where 𝑋𝑈, yields degree 𝑃 + 1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑈𝑉    &   𝐼 = (iEdg‘𝐺)    &   𝐼 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}    &   ((VtxDeg‘𝐺)‘𝑈) = 𝑃    &   (Vtx‘𝐹) = 𝑉    &   𝑋𝑉    &   𝑋𝑈    &   (iEdg‘𝐹) = (𝐼 ++ ⟨“{𝑋, 𝑈}”⟩)       ((VtxDeg‘𝐹)‘𝑈) = (𝑃 + 1)

Theoremvtxdginducedm1lem1 26491 Lemma 1 for vtxdginducedm1 26495: the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       (iEdg‘𝑆) = 𝑃

Theoremvtxdginducedm1lem2 26492* Lemma 2 for vtxdginducedm1 26495: the domain of the edge function in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       dom (iEdg‘𝑆) = 𝐼

Theoremvtxdginducedm1lem3 26493* Lemma 3 for vtxdginducedm1 26495: an edge in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁. (Contributed by AV, 16-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃       (𝐻𝐼 → ((iEdg‘𝑆)‘𝐻) = (𝐸𝐻))

Theoremvtxdginducedm1lem4 26494* Lemma 4 for vtxdginducedm1 26495. (Contributed by AV, 17-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (𝑊 ∈ (𝑉 ∖ {𝑁}) → (#‘{𝑘𝐽 ∣ (𝐸𝑘) = {𝑊}}) = 0)

Theoremvtxdginducedm1 26495* The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 17-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) +𝑒 (#‘{𝑙𝐽𝑣 ∈ (𝐸𝑙)}))

Theoremvtxdginducedm1fi 26496* The degree of a vertex 𝑣 in the induced subgraph 𝑆 of a pseudograph 𝐺 of finite size obtained by removing one vertex 𝑁 plus the number of edges joining the vertex 𝑣 and the vertex 𝑁 is the degree of the vertex 𝑣 in the pseudograph 𝐺. (Contributed by AV, 18-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (𝐸 ∈ Fin → ∀𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) = (((VtxDeg‘𝑆)‘𝑣) + (#‘{𝑙𝐽𝑣 ∈ (𝐸𝑙)})))

Theoremfinsumvtxdg2ssteplem1 26497* Lemma for finsumvtxdg2sstep 26501. (Contributed by AV, 15-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (#‘𝐸) = ((#‘𝑃) + (#‘𝐽)))

Theoremfinsumvtxdg2ssteplem2 26498* Lemma for finsumvtxdg2sstep 26501. (Contributed by AV, 12-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → ((VtxDeg‘𝐺)‘𝑁) = ((#‘𝐽) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})))

Theoremfinsumvtxdg2ssteplem3 26499* Lemma for finsumvtxdg2sstep 26501. (Contributed by AV, 19-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       (((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})(#‘{𝑖𝐽𝑣 ∈ (𝐸𝑖)}) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}})) = (#‘𝐽))

Theoremfinsumvtxdg2ssteplem4 26500* Lemma for finsumvtxdg2sstep 26501. (Contributed by AV, 12-Dec-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (iEdg‘𝐺)    &   𝐾 = (𝑉 ∖ {𝑁})    &   𝐼 = {𝑖 ∈ dom 𝐸𝑁 ∉ (𝐸𝑖)}    &   𝑃 = (𝐸𝐼)    &   𝑆 = ⟨𝐾, 𝑃    &   𝐽 = {𝑖 ∈ dom 𝐸𝑁 ∈ (𝐸𝑖)}       ((((𝐺 ∈ UPGraph ∧ 𝑁𝑉) ∧ (𝑉 ∈ Fin ∧ 𝐸 ∈ Fin)) ∧ Σ𝑣𝐾 ((VtxDeg‘𝑆)‘𝑣) = (2 · (#‘𝑃))) → (Σ𝑣 ∈ (𝑉 ∖ {𝑁})((VtxDeg‘𝐺)‘𝑣) + ((#‘𝐽) + (#‘{𝑖 ∈ dom 𝐸 ∣ (𝐸𝑖) = {𝑁}}))) = (2 · ((#‘𝑃) + (#‘𝐽))))

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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 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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