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Theorem List for Metamath Proof Explorer - 25801-25900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremuhgriedg0edg0 25801 A hypergraph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.) (Proof shortened by AV, 15-Dec-2020.)
(𝐺 ∈ UHGraph → ((Edg‘𝐺) = ∅ ↔ (iEdg‘𝐺) = ∅))

Theoremuhgredgn0 25802 An edge of a hypergraph is a nonempty subset of vertices. (Contributed by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremedguhgr 25803 An edge of a hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 28-Nov-2020.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → 𝐸 ∈ 𝒫 (Vtx‘𝐺))

Theoremuhgredgrnv 25804 An edge of a hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 4-Jun-2021.)
((𝐺 ∈ UHGraph ∧ 𝐸 ∈ (Edg‘𝐺) ∧ 𝑁𝐸) → 𝑁 ∈ (Vtx‘𝐺))

Theoremuhgredgss 25805 The set of edges of a hypergraph is a subset of the powerset of vertices without the empty set. (Contributed by AV, 29-Nov-2020.)
(𝐺 ∈ UHGraph → (Edg‘𝐺) ⊆ (𝒫 (Vtx‘𝐺) ∖ {∅}))

Theoremupgredgss 25806* The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 29-Nov-2020.)
(𝐺 ∈ UPGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgredgss 25807* The set of edges of a multigraph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 25-Nov-2020.)
(𝐺 ∈ UMGraph → (Edg‘𝐺) ⊆ {𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (#‘𝑥) = 2})

Theoremedgupgr 25808 Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (#‘𝐸) ≤ 2))

Theoremedgumgr 25809 Properties of an edge of a multigraph. (Contributed by AV, 25-Nov-2020.)
((𝐺 ∈ UMGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ (#‘𝐸) = 2))

Theoremuhgrvtxedgiedgb 25810* In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))

Theoremupgredg 25811* For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})

Theoremumgredg 25812* For each edge in a multigraph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 25-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 (𝑎𝑏𝐶 = {𝑎, 𝑏}))

Theoremumgrpredgav 25813 An edge of a multigraph always connects two vertices. Analogue of umgredgprv 25773. This theorem does not hold for arbitrary pseudographs: if either 𝑀 or 𝑁 is a proper class, then {𝑀, 𝑁} ∈ 𝐸 could still hold ({𝑀, 𝑁} would be either {𝑀} or {𝑁}, see prprc1 4243 or prprc2 4244, i.e. a loop), but 𝑀𝑉 or 𝑁𝑉 would not be true. (Contributed by AV, 27-Nov-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → (𝑀𝑉𝑁𝑉))

Theoremupgredg2vtx 25814* For a vertex incident to an edge there is another vertex incident to the edge in a pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 5-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐶𝐸𝐴𝐶) → ∃𝑏𝑉 𝐶 = {𝐴, 𝑏})

Theoremupgredgpr 25815 If a proper pair (of vertices) is a subset of an edge in a pseudograph, the pair is the edge. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝐺 ∈ UPGraph ∧ 𝐶𝐸 ∧ {𝐴, 𝐵} ⊆ 𝐶) ∧ (𝐴𝑈𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} = 𝐶)

Theoremumgredgne 25816 An edge of a multigraph always connects two different vertices. Analog of umgrnloopv 25772 resp. umgrnloop 25774. (Contributed by AV, 27-Nov-2020.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ {𝑀, 𝑁} ∈ 𝐸) → 𝑀𝑁)

Theoremumgrnloop2 25817 A multigraph has no loops. (Contributed by AV, 27-Oct-2020.) (Revised by AV, 30-Nov-2020.)
(𝐺 ∈ UMGraph → {𝑁, 𝑁} ∉ (Edg‘𝐺))

Theoremumgredgnlp 25818* An edge of a multigraph is not a loop. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 8-Jun-2021.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UMGraph ∧ 𝐶𝐸) → ¬ ∃𝑣 𝐶 = {𝑣})

PART 17  GRAPH THEORY (DEPRECATED)

To give an overview of the definitions and terms used in the context of graph theory, a glossary is provided in the following, mainly according to definitions in [Bollobas] p. 1-8 or in [Diestel] p. 2-28. Although this glossary concentrates on undirected graphs, many of the concepts are also useful for directed graphs.

Basic kinds of graphs:

TermReferenceDefinitionRemarks
Undirected hypergraph df-uhgra 25821 an ordered pair 𝑉, 𝐸 of a set 𝑉 and a function 𝐸 into the powerset of 𝑉 (ran 𝐸 ⊆ (𝒫 𝑉)).
An element of 𝑉 is called "vertex", an element of ran 𝐸 is called "edge", the function 𝐸 is called the "edge-function" .
In this most general definition of a graph, an "edge" may connect three or more vertices with each other, see [Berge] p. 1.
In Wikipedia "Hypergraph", see https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020) such a hypergraph is called "non-simple hypergraph", "multiple hypergraph" or "multi-hypergraphs". According to Wikipedia "Incidence structure", see https://en.wikipedia.org/wiki/Incidence_structure (18-Jan-2020) "Each hypergraph [...] can be regarded as an incidence structure in which the [vertices] play the role of "points", the corresponding family of [edges] plays the role of "lines" and the incidence relation is set membership".

If a graph is represented by a class variable, e.g. 𝐺, the edges of this graph are often represented by the function value (Edges‘𝐺). If the graph is given as pair 𝑉, 𝐸, however, (Edges‘⟨𝑉, 𝐸⟩) or preferably (𝑉Edges𝐸) is only used to talk about edges more explicitly. Otherwise, ran 𝐸 is used, because this is much shorter.
Notice that by using (Edges‘𝐺) the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this representation will only be used for undirected simple graphs.
For the set of vertices, a function "Vertices" could have been defined analogously. But "( Vertices ` G )" would have been exactly the same as (1st𝐺), so the latter is used to denote the set of vertices if the graph is represented by a class variable.
Undirected simple hypergraph df-ushgra 25822 an ordered pair 𝑉, 𝐸 of a set 𝑉 and a one-to-one function 𝐸 into the powerset of 𝑉 (ran 𝐸 ⊆ (𝒫 𝑉)). See also Wikipedia "Hypergraph", https://en.wikipedia.org/wiki/Hypergraph (18-Jan-2020). This is how a "hypergraph" is defined in Section I.1 in [Bollobas] p. 7 or the definition in section 1.10 in [Diestel] p. 27. A simple hypergraph has at most one edge between the same vertices, hence a multigraph needs not to be a simple hypergraph.
According to [Berge] p. 1, "A simple hypergraph (or "Sperner family") is a hypergraph H = { E1, E2, ..., Em } such that Ei C_ Ej => i = j". By this definition, a simple hypergraph cannot contain the edges E1 = { v1 , v2 } and E2 = { v1, v2, v3 }, because E1 C_ E2, but 1 =/= 2.
Undirected multigraph df-umgra 25842 a graph 𝑉, 𝐸 such that 𝐸 is a function into the set of (proper or not proper) unordered pairs of 𝑉.A proper unordered pair contains two different elements, a not proper unordered pair contains two times the same element, so it is a singleton (see preqsn 4331).
According to the definition in Section I.1 in [Bollobas] p. 7, "In a multigraph both multiple edges [joining two vertices] and multiple loops [joining a vertex to itself] are allowed", or according to [Diestel] p. 28, "A multigraph is a pair (V,E) of disjoint sets (of vertices and edges) together with a map E -> V u. [V]^2 assigning to every edge either one or two vertices, its end.".
Undirected simple graph with loops df-uslgra 25861 a graph 𝑉, 𝐸 such that 𝐸 is a one-to-one function into the set of (proper or not proper) unordered pairs of 𝑉.This means that there is at most one edge between two vertices, and at most one loop from a vertex to itself.
Undirected simple graph without loops (in short "simple graph") df-usgra 25862 a graph 𝑉, 𝐸 such that 𝐸 is a one-to-one function into the set of (proper) unordered pairs of 𝑉.An ordered pair 𝑉, 𝐸 of two distinct sets 𝑉 and 𝐸 (the "usual" definition of a "graph", see, for example, the definition in section I.1 of [Bollobas] p. 1 or in section 1.1 of [Diestel] p. 2) can be identified with an undirected simple graph without loops by "indexing" the edges with themselves, see ausisusgra 25884.
Finite graph---a graph 𝑉, 𝐸 with finite sets 𝑉 and 𝐸.See definitions in [Bollobas] p. 1 or [Diestel] p. 2.
In simple graphs, 𝐸 is finite if 𝑉 is finite, see usgrafis 25944. The number of edges is limited by (𝑛C2) (or "𝑛 choose 2") with 𝑛 = (#‘𝑉), see usgramaxsize 26015. Analogously, the number of edges of an undirected simple graph with loops is limited by ((𝑛 + 1)C2). In multigraphs, however, 𝐸 can be infinite although 𝑉 is finite.
Graph of finite size---a graph 𝑉, 𝐸 with finite set 𝐸, i.e. with a finite number of edges.A graph can be of finite size although 𝑉 is infinite.

Terms and properties of graphs:
TermReferenceDefinitionRemarks
Edge joining (two) vertices --- An edge 𝑒 ∈ ran 𝐸 "joins" the vertices v1, v2, ... vn (𝑛 ∈ ℕ) if 𝑒 = { v1, v2, ... vn }. If 𝑛 = 1, 𝑒 = { v1 } is a "loop", if 𝑛 = 2, 𝑒 = { v1 , v2 } is an edge as it is usually defined, see definition in Section I.1 in [Bollobas] p. 1.
(Two) Endvertices of an edge see definition in Section I.1 in [Bollobas] p. 1. If an edge 𝑒 ∈ ran 𝐸 joins the vertices v1, v2, ... vn (𝑛 ∈ ℕ), then the vertices v1, v2, ... vn are called the "endvertices" of the edge 𝑒.
(Two) Adjacent vertices see definition in Section I.1 in [Bollobas] p. 1/2. The vertices v1, v2, ... vn (𝑛 ∈ ℕ) are "adjacent" if there is an edge e = { v1, v2, ... vn } joining these vertices. In this case, the vertices are "incident" with the edge e (see definition in Section I.1 in [Bollobas] p. 2) or "connected" by the edge e.
Edge ending at a vertex An edge 𝑒 ∈ ran 𝐸 is "ending" at a vertex 𝑣 if the vertex is an endvertex of the edge: 𝑣𝑒. In other words, the vertex 𝑣 is incident with the edge 𝑒.
(Two) Adjacent edges The edges e0, e1, ... en (𝑛 ∈ ℕ) are "adjacent" if they have exactly one common endvertex. Generalization of definition in Section I.1 in [Bollobas] p. 2.
Order of a graph see definition in Section I.1 in [Bollobas] p. 3 the "order" of a graph 𝑉, 𝐸 is the number of vertices in the graph ((#‘𝑉)).
Size of a graph see definition in Section I.1 in [Bollobas] p. 3 the "size" of a graph 𝑉, 𝐸 is the number of edges in the graph ((#‘𝐸)). Or, for simple graphs 𝐺: (#‘(Edges‘𝐺))).
Neighborhood of a vertex df-nbgra 25949 resp. definition in Section I.1 in [Bollobas] p. 3 A vertex connected with a vertex 𝑣 by an edge is called a "neighbor" of the vertex 𝑣. The set of neighbors of a vertex 𝑣 is called the "neighborhood" (or "open neighborhood") of the vertex 𝑣. The "closed neighborhood" is the union of the (open) neighborhood of the vertex 𝑣 with {𝑣}.
Degree of a vertex df-vdgr 26421 The "degree" of a vertex is the number of the edges ending at this vertex. In a simple graph, the degree of a vertex is the number of neighbors of this vertex, see definition in Section I.1 in [Bollobas] p. 3
Isolated vertex usgravd0nedg 26445 A vertex is called "isolated" if it is not an endvertex of any edge, thus having degree 0.
Universal vertex df-uvtx 25951 A vertex is called "universal" if it is connected with every other vertex of the graph by an edge, thus having degree (#‘𝑉).

Special kinds of graphs:
TermReferenceDefinitionRemarks
Complete graph df-cusgra 25950 A graph is called "complete" if each pair of vertices is connected by an edge. The size of a complete undirected simple graph of order 𝑛 is (𝑛C2) (or "𝑛 choose 2"), see cusgrasize 26006.
Empty graph umgra0 25854 and usgra0 25899 A graph is called "empty" if it has no edges.
Null graph usgra0v 25900 A graph is called the "null graph" if it has no vertices (and therefore also no edges).
Trivial graph usgra1v 25919 A graph is called the "trivial graph" if it has only one vertex and no edges.
Connected graph df-conngra 26198 resp. definition in Section I.1 in [Bollobas] p. 6 A graph is called "connected" if for each pair of vertices there is a path between these vertices.

For the terms "Path", "Walk", "Trail", "Circuit", "Cycle" see the remarks below and the definitions in Section I.1 in [Bollobas] p. 4-5.

17.1  Undirected graphs - basics

17.1.1  Undirected hypergraphs

Syntaxcuhg 25819 Extend class notation with undirected hypergraphs.
class UHGrph

Syntaxcushg 25820 Extend class notation with undirected simple hypergraphs.
class USHGrph

Definitiondf-uhgra 25821* Define the class of all undirected hypergraphs. An undirected hypergraph is a pair of a set and a function into the powerset of this set (the empty set excluded). (Contributed by Alexander van der Vekens, 26-Dec-2017.)
UHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}

Definitiondf-ushgra 25822* Define the class of all undirected simple hypergraphs. An undirected simple hypergraph is a pair of a set and an injective (one-to-one) function into the powerset of this set (the empty set excluded). (Contributed by AV, 19-Jan-2020.)
USHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→(𝒫 𝑣 ∖ {∅})}

Theoremreluhgra 25823 The class of all undirected hypergraphs is a relation. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
Rel UHGrph

Theoremrelushgra 25824 The class of all undirected simple hypergraphs is a relation. (Contributed by AV, 19-Jan-2020.)
Rel USHGrph

Theoremuhgrav 25825 The classes of vertices and edges of an undirected hypergraph are sets. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))

Theoremuhgraopelvv 25826 An undirected hypergraph is a member in the universal class of ordered pairs. (Contributed by AV, 3-Jan-2020.)
(𝐺 ∈ UHGrph → 𝐺 ∈ (V × V))

Theoremisuhgra 25827 The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
((𝑉𝑊𝐸𝑋) → (𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgraf 25828 The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))

Theoremuhgrafun 25829 The edge function of an undirected hypergraph is a function. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(𝑉 UHGrph 𝐸 → Fun 𝐸)

Theoremisushgra 25830 The property of being an undirected simple hypergraph. (Contributed by AV, 3-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (𝑉 USHGrph 𝐸𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))

Theoremushgraf 25831 The edge function of an undirected simple hypergraph is a function into the power set of the set of vertices. (Contributed by AV, 19-Jan-2020.)
(𝑉 USHGrph 𝐸𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Theoremushgrauhgra 25832 An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.)
(𝑉 USHGrph 𝐸𝑉 UHGrph 𝐸)

Theoremuhgraop 25833 The property of being an undirected hypergraph represented as an ordered pair. The representation as an ordered pair is the usual representation of a graph, see section I.1 of [Bollobas] p. 1. (Contributed by AV, 1-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (⟨𝑉, 𝐸⟩ ∈ UHGrph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Theoremuhgrac 25834 The property of being an undirected hypergraph represented by a class. This representation is useful if the set of vertices and the edge function is/needs not to be known. (Contributed by AV, 1-Jan-2020.)
(𝐺 ∈ UHGrph → (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅}))

Theoremuhgrass 25835 An edge is a subset of vertices, analogous to umgrass 25848. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
((𝑉 UHGrph 𝐸𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremuhgraeq12d 25836 Equality of hypergraphs. (Contributed by Alexander van der Vekens, 26-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 UHGrph 𝐸𝑊 UHGrph 𝐹))

Theoremuhgrares 25837 A subgraph of a hypergraph (formed by removing some edges from the original graph) is a hypergraph, analogous to umgrares 25853. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉 UHGrph 𝐸𝑉 UHGrph (𝐸𝐴))

Theoremuhgra0 25838 The empty graph, with vertices but no edges, is a hypergraph, analogous to umgra0 25854. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉𝑊𝑉 UHGrph ∅)

Theoremuhgra0v 25839 The null graph, with no vertices, is a hypergraph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(∅ UHGrph 𝐸𝐸 = ∅)

Theoremuhgraun 25840 The union of two (undirected) hypergraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are hypergraphs, then 𝑉, 𝐸𝐹 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices), analogous to umgraun 25857. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UHGrph 𝐸)    &   (𝜑𝑉 UHGrph 𝐹)       (𝜑𝑉 UHGrph (𝐸𝐹))

17.1.2  Undirected multigraphs

Syntaxcumg 25841 Extend class notation with undirected multigraphs.
class UMGrph

Definitiondf-umgra 25842* Define the class of all undirected multigraphs. A multigraph is a pair 𝑉, 𝐸 where 𝐸 is a function into subsets of 𝑉 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. (Contributed by Mario Carneiro, 11-Mar-2015.)
UMGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Theoremrelumgra 25843 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
Rel UMGrph

Theoremisumgra 25844* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉𝑊𝐸𝑋) → (𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremwrdumgra 25845* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉𝑊𝐸 ∈ Word 𝑋) → (𝑉 UMGrph 𝐸𝐸 ∈ Word {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremumgraf2 25846* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgraf 25847 without explicitly specified domain of the edge function. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉 UMGrph 𝐸𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgraf 25847* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremumgrass 25848 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ⊆ 𝑉)

Theoremumgran0 25849 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Theoremumgrale 25850 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (#‘(𝐸𝐹)) ≤ 2)

Theoremumgrafi 25851 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ Fin)

Theoremumgraex 25852* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
((𝑉 UMGrph 𝐸𝐸 Fn 𝐴𝐹𝐴) → ∃𝑥𝑉𝑦𝑉 (𝐸𝐹) = {𝑥, 𝑦})

Theoremumgrares 25853 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉 UMGrph 𝐸𝑉 UMGrph (𝐸𝐴))

Theoremumgra0 25854 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝑉𝑊𝑉 UMGrph ∅)

Theoremumgra1 25855 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
(((𝑉𝑊𝐴𝑋) ∧ (𝐵𝑉𝐶𝑉)) → 𝑉 UMGrph {⟨𝐴, {𝐵, 𝐶}⟩})

Theoremumisuhgra 25856 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.)
(𝑉 UMGrph 𝐸𝑉 UHGrph 𝐸)

Theoremumgraun 25857 The union of two (undirected) multigraphs (with the same vertex set): If 𝑉, 𝐸 and 𝑉, 𝐹 are graphs, then 𝑉, 𝐸𝐹 is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
(𝜑𝐸 Fn 𝐴)    &   (𝜑𝐹 Fn 𝐵)    &   (𝜑 → (𝐴𝐵) = ∅)    &   (𝜑𝑉 UMGrph 𝐸)    &   (𝜑𝑉 UMGrph 𝐹)       (𝜑𝑉 UMGrph (𝐸𝐹))

17.1.3  Undirected simple graphs

17.1.3.1  Undirected simple graphs - basics

Syntaxcuslg 25858 Extend class notation with undirected (simple) graphs with loops.
class USLGrph

Syntaxcusg 25859 Extend class notation with undirected (simple) graphs (without loops).
class USGrph

Syntaxcedg 25860 Extend class notation with the set of edges (of an undirected simple graph).
class Edges

Definitiondf-uslgra 25861* Define the class of all undirected simple graphs with loops. An undirected simple graph with loops is a special undirected multigraph 𝑉, 𝐸 where 𝐸 is an injective (one-to-one) function into subsets of 𝑉 of cardinality one or two, representing the two vertices incident to the edge, or the one vertex if the edge is a loop. In contrast to a multigraph, there is at most one edge between two vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USLGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}

Definitiondf-usgra 25862* Define the class of all undirected simple graphs without loops. An undirected simple graph without loops is a special undirected simple graph 𝑉, 𝐸 where 𝐸 is an injective (one-to-one) function into subsets of 𝑉 of cardinality two, representing the two vertices incident to the edge. Such graphs are usually simply called "undirected graphs", so if only the term "undirected graph" is used, an undirected simple graph without loops is meant. Therefore, an undirected graph has no loops (edges a vertex to itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
USGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) = 2}}

Theoremreluslgra 25863 The class of all undirected simple graph with loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Rel USLGrph

Theoremrelusgra 25864 The class of all undirected simple graph without loops is a relation. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Rel USGrph

Definitiondf-edg 25865 Define the class of edges of a graph, see also definition ("E = E(G)") in section I.1 of [Bollobas] p. 1. This definition is very general: It defines edges for any ordered pairs as the range of its second component (which even needs not to be a function). Therefore, this definition could also be used for hypergraphs and multigraphs. In these cases, however, the (possibly more than one) edges connecting the same vertices could not be distinguished anymore. Therefore, this definition should only be used for undirected simple graphs. (Contributed by AV, 1-Jan-2020.)
Edges = (𝑔 ∈ V ↦ ran (2nd𝑔))

Theoremuslgrav 25866 The classes of vertices and edges of an undirected simple graph with loops are sets. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
(𝑉 USLGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))

Theoremusgrav 25867 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
(𝑉 USGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))

Theoremedgval 25868 The edges of a graph. (Contributed by AV, 1-Jan-2020.)
(𝐺𝑉 → (Edges‘𝐺) = ran (2nd𝐺))

Theoremedgopval 25869 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (Edges‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Theoremedgov 25870 The edges of a graph, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 25869. The representation ran 𝐸 for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.)
((𝑉𝑊𝐸𝑋) → (𝑉Edges𝐸) = ran 𝐸)

Theoremedguslgra 25871 The edges of an undirected simple graph with loops. (Contributed by AV, 2-Jan-2020.)
(𝑉 USLGrph 𝐸 → (𝑉Edges𝐸) = ran 𝐸)

Theoremisuslgra 25872* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
((𝑉𝑊𝐸𝑋) → (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))

Theoremisusgra 25873* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
((𝑉𝑊𝐸𝑋) → (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2}))

Theoremuslgraf 25874* The edge function of an undirected simple graph with loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})

Theoremusgraf 25875* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) = 2})

Theoremisusgra0 25876* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
((𝑉𝑊𝐸𝑋) → (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremusgraf0 25877* The edge function of an undirected simple graph without loops is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
(𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})

Theoremusgrafun 25878 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
(𝑉 USGrph 𝐸 → Fun 𝐸)

Theoremusgraop 25879* An undirected simple graph without loops represented as ordered pair with a one-to-one edge function. (Contributed by AV, 1-Jan-2020.)
(𝐺 ∈ USGrph → ∃𝑣𝑒(𝐺 = ⟨𝑣, 𝑒⟩ ∧ 𝑒:dom 𝑒1-1→{𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}))

Theoremusgrac 25880 An undirected simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.)
(𝐺 ∈ USGrph → (1st𝐺) USGrph (2nd𝐺))

Theoremedgss 25881* The set of edges of an undirected simple graph without loops is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.)
(𝐺 ∈ USGrph → (Edges‘𝐺) ⊆ {𝑥 ∈ 𝒫 (1st𝐺) ∣ (#‘𝑥) = 2})

Theoremedg 25882 An edge of an undirected simple graph without loops is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.)
((𝐺 ∈ USGrph ∧ 𝐸 ∈ (Edges‘𝐺)) → (𝐸 ∈ 𝒫 (1st𝐺) ∧ (#‘𝐸) = 2))

Theoremisausgra 25883* The property of an unordered pair to be an alternatively defined undirected simple graph without loops (defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))

Theoremausisusgra 25884* The equivalence of the definitions of an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 28-Aug-2017.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑋𝐸𝑌) → (𝑉𝐺𝐸𝑉 USGrph ( I ↾ 𝐸)))

Theoremausisusgraedg 25885* The equivalence of the definitions of an undirected simple graph without loops, expressed with the set of edges. (Contributed by AV, 2-Jan-2020.)
𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}       ((𝑉𝑊𝐸𝑋) → (𝑉𝐺(𝑉Edges𝐸) ↔ 𝑉 USGrph ( I ↾ ran 𝐸)))

Theoremusgraedgop 25886 An edge of an undirected simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
((𝑉 USGrph 𝐸𝑋 ∈ dom 𝐸) → ((𝐸𝑋) = {𝑀, 𝑁} ↔ ⟨𝑋, {𝑀, 𝑁}⟩ ∈ 𝐸))

Theoremusgraf1o 25887 The edge function of an undirected simple graph without loops is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
(𝑉 USGrph 𝐸𝐸:dom 𝐸1-1-onto→ran 𝐸)

Theoremuslgraf1oedg 25888 The edge function of an undirected simple graph with loops is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.)
(𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1-onto→(𝑉Edges𝐸))

Theoremusgraf1 25889 The edge function of an undirected simple graph without loops is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
(𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)

Theoremusgrass 25890 An edge is a subset of vertices, analogous to umgrass 25848. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
((𝑉 USGrph 𝐸𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Theoremusgraeq12d 25891 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑉 = 𝑊𝐸 = 𝐹)) → (𝑉 USGrph 𝐸𝑊 USGrph 𝐹))

Theoremuslisushgra 25892 An undirected simple graph with loops is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.)
(𝑉 USLGrph 𝐸𝑉 USHGrph 𝐸)

Theoremuslisumgra 25893 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(𝑉 USLGrph 𝐸𝑉 UMGrph 𝐸)

Theoremusisuslgra 25894 An undirected simple graph without loops is an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.)
(𝑉 USGrph 𝐸𝑉 USLGrph 𝐸)

Theoremusisumgra 25895 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
(𝑉 USGrph 𝐸𝑉 UMGrph 𝐸)

Theoremusisuhgra 25896 An undirected simple graph without loops is an undirected hypergraph. (Contributed by Alexander van der Vekens, 9-Feb-2018.)
(𝑉 USGrph 𝐸𝑉 UHGrph 𝐸)

Theoremelusuhgra 25897 An undirected simple graph without loops is an undirected hypergraph. (Contributed by AV, 9-Jan-2020.)
(𝐺 ∈ USGrph → 𝐺 ∈ UHGrph )

Theoremusgrares 25898 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 25853. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
(𝑉 USGrph 𝐸𝑉 USGrph (𝐸𝐴))

Theoremusgra0 25899 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 25854. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Proof shortened by AV, 25-Nov-2020.)
(𝑉𝑊𝑉 USGrph ∅)

Theoremusgra0v 25900 The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
(∅ USGrph 𝐸𝐸 = ∅)

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