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Theorem List for Metamath Proof Explorer - 40701-40800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvtxdumgrval 40701* 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 40702* 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 40703* 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 40704* Lemma for vtxdushgrfvedg 40705 and vtxdusgrfvedg 40706. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.) TODO-AV: proof can be shortened by using "bj-eleq2w", after it is moved to main.set.
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ USHGraph ∧ 𝑈𝑉) → (#‘{𝑖 ∈ dom (iEdg‘𝐺) ∣ 𝑈 ∈ ((iEdg‘𝐺)‘𝑖)}) = (#‘{𝑒𝐸𝑈𝑒}))

Theoremvtxdushgrfvedg 40705* 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 40706* 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 40707* 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 40708* 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 40709* 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 40710* 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 40711* Alternate proof of vtxdusgr0edgnel 40710, not based on vtxduhgr0edgnel 40709. 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 40712 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 40713* 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 40714* In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. Formerly usgn0fidegnn0 26556. (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 40715 A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 40475). (Contributed by AV, 21-Feb-2021.)
(𝜑 → (Vtx‘𝐺) = 𝑉)    &   (𝜑𝐴𝑋)    &   (𝜑𝑁𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})       (𝜑𝐺 ∈ USPGraph )

Theorem1loopgredg 40716 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 40717 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 40718 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 40719 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 40720 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 40721 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)

Theorem1hegrlfgr 40722* --- TODO-AV: not used anymore!? ! ------------------------- A graph 𝐺 with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
(𝜑𝐴𝑋)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐵𝐶)    &   (𝜑𝐸 ∈ 𝒫 𝑉)    &   (𝜑 → (iEdg‘𝐺) = {⟨𝐴, 𝐸⟩})    &   (𝜑 → {𝐵, 𝐶} ⊆ 𝐸)       (𝜑 → (iEdg‘𝐺):{𝐴}⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (#‘𝑥)})

Theorem1hegrvtxdg1 40723 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 40724 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 40725 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 40726 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 40727 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 40728 Lemma for p1evtxdeq 40729 and p1evtxdp1 40730. (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 40729 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 40730 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 40731 The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       (𝑉𝑊 → (Vtx‘𝐺) = 𝑉)

Theoremuspgrloopvtxel 40732 A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475). (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝑁𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuspgrloopiedg 40733 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (iEdg‘𝐺) = {⟨𝐴, {𝑁}⟩})

Theoremuspgrloopedg 40734 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩       ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Theoremuspgrloopnb0 40735 In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475), 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 40736 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 40475), 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 40737 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 40738 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 40739 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 40740 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 40741 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 40742 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 40743 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 40744 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 40735, but degree 2, see uspgrloopvd2 40736. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 40742, but also degree 2, see umgr2v2evd2 40743. (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 40745 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 40746* 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 40747* 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 40748* The degree of a vertex in a simple graphs 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 40749* 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 40750* 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 40751* 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-av 40752* 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-av 40753* The induction step for a vertex degree calculation. 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-av 40754* The induction step for a vertex degree calculation. 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)

21.34.8.8  Regular graphs

With df-rgr 40757 and df-rusgr 40758, k-regularity of a (simple) graph is defined as predicate RegGraph resp. RegUSGraph. Instead of defining a predicate, an alternative could have been to define a function that maps an extended nonnegative integer to the class of "graphs" in which every vertex has the extended nonnegative integer as degree: RegGraph = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘}). This function, however, would not be defined for 𝑘 = 0 (see rgrx0nd 40794), because {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} is not a set (see rgrprcx 40792). It is expected that this function is not defined for every 𝑘 ∈ ℕ0* (how could this be proven?).

Syntaxcrgr 40755 Extend class notation to include the class of all regular graphs.
class RegGraph

Syntaxcrusgr 40756 Extend class notation to include the class of all regular simple graphs.
class RegUSGraph

Definitiondf-rgr 40757* Define the "k-regular" predicate, which is true for a "graph" being k-regular: read 𝐺 RegGraph 𝐾 as "𝐺 is 𝐾-regular" or "𝐺 is a 𝐾-regular graph". Note that 𝐾 is allowed to be positive infinity (𝐾 ∈ ℕ0*), as proposed by GL. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 26-Dec-2020.)
RegGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑘 ∈ ℕ0* ∧ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘)}

Definitiondf-rusgr 40758* Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read 𝐺 RegUSGraph 𝐾 as 𝐺 is a 𝐾-regular simple graph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 18-Dec-2020.)
RegUSGraph = {⟨𝑔, 𝑘⟩ ∣ (𝑔 ∈ USGraph ∧ 𝑔 RegGraph 𝑘)}

Theoremisrgr 40759* The property of a class being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺𝑊𝐾𝑍) → (𝐺 RegGraph 𝐾 ↔ (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))

Theoremrgrprop 40760* The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 RegGraph 𝐾 → (𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))

Theoremisrusgr 40761 The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.)
((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾)))

Theoremrusgrprop 40762 The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
(𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐺 RegGraph 𝐾))

Theoremrusgrrgr 40763 A k-regular simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
(𝐺 RegUSGraph 𝐾𝐺 RegGraph 𝐾)

Theoremrusgrusgr 40764 A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
(𝐺 RegUSGraph 𝐾𝐺 ∈ USGraph )

Theoremfinrusgrfusgr 40765 A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) → 𝐺 ∈ FinUSGraph )

Theoremisrusgr0 40766* The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺𝑊𝐾𝑍) → (𝐺 RegUSGraph 𝐾 ↔ (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾)))

Theoremrusgrprop0 40767* The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾))

Theoremusgreqdrusgr 40768* If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (𝐷𝑣) = 𝐾) → 𝐺 RegUSGraph 𝐾)

Theoremfusgrregdegfi 40769* In a nonempty finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 19-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾𝐾 ∈ ℕ0))

Theoremfusgrn0eqdrusgr 40770* If all vertices in a nonempty finite simple graph have the same (finite) degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 (𝐷𝑣) = 𝐾𝐺 RegUSGraph 𝐾))

Theoremfrusgrnn0 40771 In a nonempty finite k-regular simple graph, the degree of each vertex is finite. (Contributed by AV, 7-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝐺 RegUSGraph 𝐾𝑉 ≠ ∅) → 𝐾 ∈ ℕ0)

Theorem0edg0rgr 40772 A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
((𝐺𝑊 ∧ (iEdg‘𝐺) = ∅) → 𝐺 RegGraph 0)

Theoremuhgr0edg0rgr 40773 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Edg‘𝐺) = ∅) → 𝐺 RegGraph 0)

Theoremuhgr0edg0rgrb 40774 A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
(𝐺 ∈ UHGraph → (𝐺 RegGraph 0 ↔ (Edg‘𝐺) = ∅))

Theoremusgr0edg0rusgr 40775 A simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 19-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
(𝐺 ∈ USGraph → (𝐺 RegUSGraph 0 ↔ (Edg‘𝐺) = ∅))

Theorem0vtxrgr 40776* A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegGraph 𝑘)

Theorem0vtxrusgr 40777* A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
((𝐺𝑊 ∧ (Vtx‘𝐺) = ∅ ∧ (iEdg‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)

Theorem0uhgrrusgr 40778* The null graph as hypergraph is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
((𝐺 ∈ UHGraph ∧ (Vtx‘𝐺) = ∅) → ∀𝑘 ∈ ℕ0* 𝐺 RegUSGraph 𝑘)

Theorem0grrusgr 40779 The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.)
𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘

Theorem0grrgr 40780 The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
𝑘 ∈ ℕ0* ∅ RegGraph 𝑘

Theoremcusgrrusgr 40781 A complete simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → 𝐺 RegUSGraph ((#‘𝑉) − 1))

Theoremcusgrm1rusgr 40782 A finite simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGraph was allowed for 𝑘 ∈ ℤ, then the assumption 𝑉 ≠ ∅ could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (𝐺 ∈ ComplUSGraph ↔ 𝐺 RegUSGraph ((#‘𝑉) − 1)))

Theoremrusgrpropnb 40783* The properties of a k-regular simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 26-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (#‘(𝐺 NeighbVtx 𝑣)) = 𝐾))

Theoremrusgrpropedg 40784* The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (#‘{𝑒 ∈ (Edg‘𝐺) ∣ 𝑣𝑒}) = 𝐾))

Theoremrusgrpropadjvtx 40785* The properties of a k-regular simple graph expressed with adjacent vertices. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 27-Dec-2020.)
𝑉 = (Vtx‘𝐺)       (𝐺 RegUSGraph 𝐾 → (𝐺 ∈ USGraph ∧ 𝐾 ∈ ℕ0* ∧ ∀𝑣𝑉 (#‘{𝑘𝑉 ∣ {𝑣, 𝑘} ∈ (Edg‘𝐺)}) = 𝐾))

Theoremrusgrnumwrdl2 40786* In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 RegUSGraph 𝐾𝑃𝑉) → (#‘{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ (𝑤‘0) = 𝑃 ∧ {(𝑤‘0), (𝑤‘1)} ∈ (Edg‘𝐺))}) = 𝐾)

Theoremrusgr1vtxlem 40787* Lemma for rusgr1vtx 40788. (Contributed by AV, 27-Dec-2020.)
(((∀𝑣𝑉 (#‘𝐴) = 𝐾 ∧ ∀𝑣𝑉 𝐴 = ∅) ∧ (𝑉𝑊 ∧ (#‘𝑉) = 1)) → 𝐾 = 0)

Theoremrusgr1vtx 40788 If a k-regular simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.)
(((#‘(Vtx‘𝐺)) = 1 ∧ 𝐺 RegUSGraph 𝐾) → 𝐾 = 0)

Theoremrgrusgrprc 40789* The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔 ∈ USGraph ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Theoremrusgrprc 40790 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegUSGraph 0} ∉ V

Theoremrgrprc 40791 The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔𝑔 RegGraph 0} ∉ V

Theoremrgrprcx 40792* The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
{𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 0} ∉ V

Theoremrgrx0ndm 40793* 0 is not in the domain of the potentially alternatively defined vertex degree function. (Contributed by AV, 28-Dec-2020.)
𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘})       0 ∉ dom 𝑅

Theoremrgrx0nd 40794* The potentially alternatively defined vertex degree function is not defined for 0. (Contributed by AV, 28-Dec-2020.)
𝑅 = (𝑘 ∈ ℕ0* ↦ {𝑔 ∣ ∀𝑣 ∈ (Vtx‘𝑔)((VtxDeg‘𝑔)‘𝑣) = 𝑘})       (𝑅‘0) = ∅

21.34.8.9  Walks

A "walk" within a graph is usually defined for simple graphs, multigraphs or even pseudographs as "alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. This definition requires the edges to connect two vertices at most (loops are also allowed: if e(i) is a loop, then x(i-1) = x(i)). For hypergraphs containing hyperedges (i.e. edges connecting more than two vertices), however, a more general definition is needed. Two approaches for a definition applicable for arbitrary hypergraphs are used in literature: "walks on the vertex level" and "walks on the edge level" (see Aksoy, Joslyn, Marrero, Praggastis, Purvine: "Hypernetwork science via high-order hypergraph walks", June 2020, https://doi.org/10.1140/epjds/s13688-020-00231-0):

"walks on the edge level": For a positive integer s, an s-walk of length k between hyperedges f and g is a sequence of hyperedges, f=e(0), e(1), ... , e(k)=g, where for j=1, ... , k, e(j-1) =/= e(j) and e(j-1) and e(j) have at least s vertices in common (according to Aksoy et al.).

"walks on the vertex level": For a positive integer s, an s-walk of length k between vertices a and b is a sequence of vertices, a=v(0), v(1), ... , v(k)=b, where for j=1, ... , k, v(j-1) and v(j) are connected by at least s edges (analogous to Aksoy et al.).

There are two imperfections for the definition for "walks on the edge level": one is that a walk of length 1 consists of two edges (or a walk of length 0 consists of one edge), whereas a walk of length 1 on the vertex level consists of two vertices and one edge (or a walk of length 0 consists of one vertex and no edge). The other is that edges, especially loops, can be traversed only once (and not repeatedly) because of the condition e(j-1) =/= e(j). The latter is avoided in the definition for EdgWalks, see df-ewlks 40799. To be compatible with the (usual) definition of walks for pseudographs, walks also suitable for arbitrary hypergraphs are defined "on the vertex level" in the following as 1Walks, see df-1wlks 40800, restricting s to 1. 1wlk1ewlk 40844 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".

Syntaxcewlks 40795 Extend class notation with s-walks "on the edge level" (of a hypergraph).
class EdgWalks

Syntaxc1wlks 40796 Extend class notation with 1-walks (of a hypergraph). TODO-AV: should be renamed to Walks after the current definition of Walks becomes obsolete.
class 1Walks

Syntaxcwlks 40797 Extend class notation with walks (of a pseudograph).
class UPWalks

Syntaxcwlkson 40798 Extend class notation with walks between two vertices (within a graph).
class WalksOn

Definitiondf-ewlks 40799* Define the set of all s-walks of edges (in a hypergraph) corresponding to s-walks "on the edge level" discussed in Aksoy et al. For an extended nonnegative integer s, an s-walk is a sequence of hyperedges, e(0), e(1), ... , e(k), where for j=1, ... , k, e(j-1) and e(j) have at least s vertices in common. In contrast to the definition in Aksoy et al., 𝑠 = 0 (a 0-walk is an arbitrary sequence of hyperedges) and 𝑠 = +∞ (then the number of common vertices of two adjacent hyperedges must be infinite) are allowed. Furthermore, it is not forbidden that adjacent hyperedges are equal. (Contributed by AV, 4-Jan-2021.)
EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))})

Definitiondf-1wlks 40800* Define the set of all 1-walks (in a hypergraph). Such 1-walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see 1wlk1walk 40843) discussed in Aksoy et al. The predicate 𝐹(1Walks‘𝐺)𝑃 can be read as "The pair 𝐹, 𝑃 represents a walk in a graph 𝐺", see also is1wlk 40813.

The condition {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝𝑘).

According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by AV, 30-Dec-2020.)

1Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})

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