Home Metamath Proof ExplorerTheorem List (p. 391 of 402) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26569) Hilbert Space Explorer (26570-28092) Users' Mathboxes (28093-40161)

Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement

21.33.8.4  Undirected multigraphs

Syntaxcumgr 39001 Extend class notation with undirected multigraphs.
UMGraph

Definitiondf-umgr 39002* Define the class of all undirected multigraphs. An (undirected) multigraph consists of a set (of "vertices") and a function (representing indexed "edges") into subsets of 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.) (Revised by AV, 10-Oct-2020.)
UMGraph Vtx iEdg

Theoremisumgr 39003* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremwrdumgr 39004* The property of being an undirected multigraph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       Word UMGraph Word

Theoremumgrf2 39005* The edge function of an undirected multigraph is a function into unordered pairs of vertices. Version of umgraf 25043 without explicitly specified domain of the edge function (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrf 39006* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrss 39007 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrn0 39008 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrle 39009 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrfi 39010 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgrex 39011* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Vtx       iEdg       UMGraph

Theoremumgr0 39012 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
iEdg        UMGraph

Theoremumgr1lem 39013* Lemma for umgr1 39014 and uspgr1 39099. (Contributed by AV, 16-Oct-2020.)

Theoremumgr1 39014 A graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 16-Oct-2020.)
Vtx                                   iEdg        UMGraph

Theoremumgr0op 39015 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 11-Oct-2020.)
UMGraph

Theoremumgr1op 39016 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 10-Oct-2020.)
UMGraph

Theoremumgr0opALT 39017 Alternate proof of umgr0op 39015, using the general theorem gropeld 38964 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of umgr0op 39015). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
UMGraph

Theoremumgr1opALT 39018 Alternate proof of umgr1op 39016, using the general theorem gropeld 38964 to transform a theorem for an arbitrary representation of a graph into a theorem for a graph represented as ordered pair. This general approach causes some overhead, which makes the proof longer than necessary (see proof of umgr1op 39016). (Contributed by AV, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
UMGraph

Theoremumgruhgr 39019 An undirected multigraph is an undirected hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 10-Oct-2020.)
UMGraph UHGraph

Theoremumgrun 39020 The union of two (undirected) multigraphs and with the same vertex set is a multigraph with the vertex and the union of the (indexed) edges. (Contributed by AV, 12-Oct-2020.)
UMGraph        UMGraph        iEdg       iEdg       Vtx       Vtx                                    Vtx        iEdg        UMGraph

Theoremumgrunop 39021 The union of two (undirected) multigraphs (with the same vertex set): If and are graphs, then is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.)
UMGraph        UMGraph        iEdg       iEdg       Vtx       Vtx                             UMGraph

21.33.8.5  Undirected simple graphs - basics

Renaming: In Chartrand, Gary; Zhang, Ping (2012). A First Course in Graph Theory. Dover. ISBN 978-0-486-48368-9, a multigraph with loops is called a "pseudograph" (section 1.4, p. 26): "In a pseudograph, not only are parallel edges permitted but an edge is also permitted to join a vertex to itself. Such an edge is called a loop."

Proposal:

* Undirected Hypergraph: UHGraph

* Undirected simple Hypergraph: USHGraph => USHGraph C_ UHGraph

* Undirected Pseudograph: UPGraph (currently UMGraph) => UPGraph C_ UHGraph

* Undirected Muligraph: UMGraph (not defined yet) => UMGraph C_ UPGraph

* Undirected simple Pseudograph: USPGraph => USPGraph C_ UPGraph => USPGraph C_ USHGraph

* Undirected simple Graph: USGraph => USGraph C_ USPGraph => USGraph C_ UMGraph

In this section, "simple graph" will always stand for "undirected simple graph (without loops)" and "simple pseudograph" for "undirected simple pseudograph (which could have loops)".

Syntaxcuspgr 39022 Extend class notation with undirected simple pseudographs (which could have loops).
USPGraph

Syntaxcusgr 39023 Extend class notation with undirected simple graphs (without loops).
USGraph

Syntaxcedga 39024 Extend class notation with the set of edges (of an undirected simple (pseudo)graph) Remark: If this definition (and all related theorems) are moved to main.set, the label should become "cedg".
Edg

Definitiondf-uspgr 39025* Define the class of all undirected simple pseudograph (which could have loops). An undirected simple pseudograph is a special undirected multigraph (see uspgrumgr 39057) or a special undirected simple hypergraph (see uspgrushgr 39056), consisting of a set (of "vertices") and an injective (one-to-one) function (representing (indexed) "edges") into subsets of 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 resp. at most one loop for a vertex. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USPGraph Vtx iEdg

Definitiondf-usgr 39026* Define the class of all undirected simple graphs (without loops). An undirected simple graph is a special undirected simple pseudograph (see usgruspgr 39058), consisting of a set (of "vertices") and an injective (one-to-one) function (representing (indexed) "edges") into subsets of of cardinality two, representing the two vertices incident to the edge. In contrast to an undirected simple pseudograph, an undirected simple graph has no loops (edges connecting a vertex with itself). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph Vtx iEdg

Definitiondf-edga 39027 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 of a class as the range of its edge function (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 (hyper)graphs (with or without loops). (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg iEdg

Theoremedgaval 39028 The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg iEdg

Theoremedgaopval 39029 The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg

Theoremedgaov 39030 The edges of a graph represented as ordered pair, 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 25065. The representation for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Edg

Theoremedgastruct 39031 The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020.)
.ef        Edg

Theoremisuspgr 39032* The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USPGraph

Theoremisusgr 39033* The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremuspgrf 39034* The edge function of a simple pseudograph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USPGraph

Theoremusgrf 39035* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremisusgr0 39036* The property of being a simple graph, simplified version of isusgr 39033. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgrf0 39037* The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 39035. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgrfun 39038 The edge function of a simple graph is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.) (Revised by AV, 13-Oct-2020.)
USGraph iEdg

Theoremusgrusgra 39039 A simple graph represented by a class induces a representation as binary relation. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Vtx USGrph iEdg

Theoremedgass 39040* The set of edges of a simple graph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Edg Vtx

Theoremuhgredg 39041 An edge of an undirected hypergraph is a subset of vertices. (Contributed by AV, 26-Oct-2020.)
UHGraph Edg Vtx

Theoremedgumgra 39042 Properties of an edge of a multigraph. (Contributed by AV, 8-Nov-2020.)
UMGraph Edg Vtx

Theoremedga 39043 An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.)
USGraph Edg Vtx

Theoremusgrop 39044 A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.)
USGraph Vtx iEdg USGraph

Theoremisausgr 39045* The property of an unordered pair to be an alternatively defined simple graph, 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.)

Theoremausgrusgrb 39046* The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.)
USGraph

Theoremusgrausgri 39047* A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.)
USGraph VtxEdg

Theoremausgrumgri 39048* If an alternatively defined simple graph has the vertices and edges of an arbitrary graph, the arbitrary graph is an undirected multigraph. (Contributed by AV, 18-Oct-2020.)
VtxEdg iEdg UMGraph

Theoremausgrusgri 39049* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 15-Oct-2020.)
VtxEdg iEdg USGraph

Theoremusgrausgrb 39050* The equivalence of the definitions of a simple graph, expressed with the set of vertices and the set of edges. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
iEdg VtxEdg USGraph

Theoremusgredgop 39051 An edge of a 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.) (Revised by AV, 15-Oct-2020.)
USGraph iEdg

Theoremusgrf1o 39052 The edge function of a simple graph is a bijective function onto its range. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       USGraph

Theoremusgrf1 39053 The edge function of a simple graph is a one to one function. (Contributed by Alexander van der Vekens, 18-Nov-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       USGraph

Theoremuspgrf1oedg 39054 The edge function of a simple graph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 15-Oct-2020.)
iEdg       USPGraph Edg

Theoremusgrss 39055 An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.)
iEdg       Vtx       USGraph

Theoremuspgrushgr 39056 A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.)
USPGraph USHGraph

Theoremuspgrumgr 39057 A simple pseudograph is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USPGraph UMGraph

Theoremusgruspgr 39058 A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USGraph USPGraph

Theoremusgruspgrb 39059* A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.)
USGraph USPGraph Edg

Theoremusgrumgr 39060 A simple graph is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.)
USGraph UMGraph

Theoremusgruhgr 39061 A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.)
USGraph UHGraph

Theoremuspgrun 39062 The union of two simple pseudographs and with the same vertex set is a multigraph with the vertex and the union of the (indexed) edges. (Contributed by AV, 16-Oct-2020.)
USPGraph        USPGraph        iEdg       iEdg       Vtx       Vtx                                    Vtx        iEdg        UMGraph

Theoremuspgrunop 39063 The union of two simple pseudographs (with the same vertex set): If and are simple pseudographs, then is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
USPGraph        USPGraph        iEdg       iEdg       Vtx       Vtx                             UMGraph

Theoremusgredg2 39064 The value of the "edge function" of a simple graph is a set containing two elements (the vertices the corresponding edge is connecting). (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.)
iEdg       USGraph

Theoremusgredgprv 39065 In a simple graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.)
iEdg       Vtx       USGraph

Theoremusgredgappr 39066 An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge) (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
Edg       USGraph

Theoremusgrpredgav 39067 An edge of a simple graph always connects two vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.)
Edg       Vtx       USGraph

Theoremedgassv2 39068 An edge of a simple graph is an unordered pair of vertices, i.e. a subset of the set of vertices of size 2. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.)
Vtx       Edg       USGraph

Theoremusgrnloopv 39069 In a simple graph, there is no loop, i.e no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.)
iEdg       USGraph

Theoremusgrnloop 39070* In a simple graph, there is no loop, i.e no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Proof shortened by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 17-Oct-2020.)
iEdg       USGraph

Theoremusgrnloop0 39071* A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.)
iEdg       USGraph

Theoremusgredgrn 39072 An edge of a simple graph always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.)
Edg       USGraph

Theoremusgrnloop1 39073 A simple graph has no loops. (Contributed by AV, 27-Oct-2020.)
USGraph Edg

Theoremusgr2edg 39074* If a vertex is adjacent to two different vertices in a simple graph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.)
iEdg       Edg       USGraph

Theoremusgr2edg1 39075* If a vertex is adjacent to two different vertices in a simple graph, there is not only one edge starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 17-Oct-2020.)
iEdg       Edg       USGraph

Theoremedgiedgb 39076* A set is an edge in a simple graph iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
iEdg       USGraph Edg

Theoremumgredg 39077* For each edge in a multigraph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
Vtx       Edg       UMGraph

Theoremusgredg 39078* For each edge in a simple graph, there are two distinct vertices which are connected by this edge. (Contributed by Alexander van der Vekens, 9-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Vtx       Edg       USGraph

Theoremusgredg3 39079* The value of the "edge function" of a simple graph is a set containing two elements (the endvertices of the corresponding edge). (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredg4 39080* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 17-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredgreu 39081* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg       USGraph

Theoremusgredg2vtx 39082* For a vertex incident to an edge there is another vertex incident to the edge. (Contributed by AV, 18-Oct-2020.)
USGraph Edg Vtx

Theoremusgredg2vtxeu 39083* For a vertex incident to an edge there is exactly one other vertex incident to the edge. (Contributed by AV, 18-Oct-2020.)
USGraph Edg Vtx

Theoremusgredg2vtxeuALT 39084* Alternate proof of usgredg2vtxeu 39083, using edgiedgb 39076, the general translation from iEdg to Edg. (Contributed by AV, 18-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
USGraph Edg Vtx

Theoremusgridx2vlem1 39085* Lemma 1 for usgraidx2v 25118. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg              USGraph

Theoremusgridx2vlem2 39086* Lemma 2 for usgraidx2v 25118. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg              USGraph

Theoremusgridx2v 39087* The mapping of indices of edges containing a given vertex into the set of vertices is 1-1. The index is mapped to the other vertex of the edge containing the vertex N. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg                     USGraph

Theoremusgredgleord 39088* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       iEdg       USGraph

Theoremiedgf1oedg 39089 The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.)
Edg       iEdg       USGraph

Theoremusgredgedga 39090* In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by by AV, 18-Oct-2020.)
Edg       iEdg       Vtx                            USGraph

Theoremusgredgaleord 39091* In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.)
Vtx       Edg       USGraph

21.33.8.6  Undirected simple graphs - examples

Theoremusgr0e 39092 The empty graph, with vertices but no edges, is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
iEdg        USGraph

Theoremusgr0vb 39093 The null graph, with no vertices, is a simple graph iff the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Revised by AV, 16-Oct-2020.)
Vtx USGraph iEdg

Theoremuhgriedg0edg0 39094 A graph has no edges iff its edge function is empty. (Contributed by AV, 21-Oct-2020.)
UHGraph Edg iEdg

Theoremuhgr0v0e 39095 The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.)
Vtx       Edg       UHGraph

Theoremuhgr0vsize0 39096 The size of a hypergraph with 0 vertices (the null graph) is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 7-Nov-2020.)
Vtx       Edg       UHGraph

Theoremusgr0v 39097 The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.)
Vtx iEdg USGraph

Theoremusgr0 39098 The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.)
USGraph

Theoremuspgr1 39099 A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.)
Vtx                                   iEdg        USPGraph

Theoremusgr1 39100 A simple graph with one edge ( with additional assumption that since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.)
Vtx                                   iEdg               USGraph

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161
 Copyright terms: Public domain < Previous  Next >