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Type | Label | Description |
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Statement | ||
Theorem | isusgr 39401* | The property of being a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.) |
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Theorem | uspgrf 39402* | 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.) |
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Theorem | usgrf 39403* | 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.) |
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Theorem | isusgrs 39404* | The property of being a simple graph, simplified version of isusgr 39401. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) (Proof shortened by AV, 24-Nov-2020.) |
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Theorem | usgrfs 39405* | The edge function of a simple graph is a one-to-one function into unordered pairs of vertices. Simplified version of usgrf 39403. (Contributed by Alexander van der Vekens, 13-Aug-2017.) (Revised by AV, 13-Oct-2020.) |
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Theorem | usgrfun 39406 | 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.) |
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Theorem | usgrusgra 39407 | 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.) |
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Theorem | usgredgss 39408* | 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.) |
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Theorem | edgusgr 39409 | An edge of a simple graph is an unordered pair of vertices. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 14-Oct-2020.) |
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Theorem | isusgrop 39410* | The property of being an undirected simple graph 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, 30-Nov-2020.) |
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Theorem | usgrop 39411 | A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020.) (Proof shortened by AV, 30-Nov-2020.) |
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Theorem | isausgr 39412* | 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.) |
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Theorem | ausgrusgrb 39413* | The equivalence of the definitions of a simple graph. (Contributed by Alexander van der Vekens, 28-Aug-2017.) (Revised by AV, 14-Oct-2020.) |
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Theorem | usgrausgri 39414* | A simple graph represented by an alternatively defined simple graph. (Contributed by AV, 15-Oct-2020.) |
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Theorem | ausgrumgri 39415* | 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.) (Revised by AV, 25-Nov-2020.) |
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Theorem | ausgrusgri 39416* | 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.) |
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Theorem | usgrausgrb 39417* | 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.) |
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Theorem | usgredgop 39418 | 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.) |
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Theorem | usgrf1o 39419 | 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.) |
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Theorem | usgrf1 39420 | 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.) |
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Theorem | uspgrf1oedg 39421 | 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.) |
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Theorem | usgrss 39422 | An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
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Theorem | uspgrushgr 39423 | A simple pseudograph is an undirected simple hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 15-Oct-2020.) |
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Theorem | uspgrupgr 39424 | A simple pseudograph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
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Theorem | uspgrupgrushgr 39425 | A graph is a simple pseudograph iff it is a pseudograph and a simple hypergraph. (Contributed by AV, 30-Nov-2020.) |
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Theorem | usgruspgr 39426 | A simple graph is a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
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Theorem | usgrumgr 39427 | A simple graph is an undirected multigraph. (Contributed by AV, 25-Nov-2020.) |
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Theorem | usgrumgruspgr 39428 | A graph is a simple graph iff it is a multigraph and a simple pseudograph. (Contributed by AV, 30-Nov-2020.) |
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Theorem | usgruspgrb 39429* | A class is a simple graph iff it is a simple pseudograph without loops. (Contributed by AV, 18-Oct-2020.) |
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Theorem | usgrupgr 39430 | A simple graph is an undirected pseudograph. (Contributed by Alexander van der Vekens, 20-Aug-2017.) (Revised by AV, 15-Oct-2020.) |
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Theorem | usgruhgr 39431 | A simple graph is an undirected hypergraph. (Contributed by AV, 9-Feb-2018.) (Revised by AV, 15-Oct-2020.) |
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Theorem | usgrislfuspgr 39432* | A simple graph is a loop-free simple pseudograph. (Contributed by AV, 27-Jan-2021.) |
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Theorem | uspgrun 39433 |
The union ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | uspgrunop 39434 |
The union of two simple pseudographs (with the same vertex set): If
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Theorem | usgrun 39435 |
The union ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | usgrunop 39436 |
The union of two simple graphs (with the same vertex set): If
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | usgredg2 39437 | 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.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | usgredg2ALT 39438 | Alternate proof of usgredg2 39437, not using umgredg2 39345. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | usgredgprv 39439 | 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.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | usgredgprvALT 39440 | Alternate proof of usgredgprv 39439, using usgredg2 39437 instead of umgredgprv 39352. (Contributed by Alexander van der Vekens, 19-Aug-2017.) (Revised by AV, 16-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | usgredgappr 39441 | An edge of a simple graph is a proper pair, i.e. a set containing two different elements (the endvertices of the edge). Analogue of usgredg2 39437. (Contributed by Alexander van der Vekens, 11-Aug-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) |
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Theorem | usgrpredgav 39442 | An edge of a simple graph always connects two vertices. Analogue of usgredgprv 39439. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 9-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Theorem | edgassv2 39443 | 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.) |
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Theorem | usgredg 39444* | 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.) (Shortened by AV, 25-Nov-2020.) |
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Theorem | usgrnloopv 39445 | 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.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | usgrnloopvALT 39446 | Alternate proof of usgrnloopv 39445, not using umgrnloopv 39351. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | usgrnloop 39447* | 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.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | usgrnloopALT 39448* | Alternate proof of usgrnloop 39447, not using umgrnloop 39353. (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.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | usgrnloop0 39449* | A simple graph has no loops. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | usgrnloop0ALT 39450* | Alternate proof of usgrnloop0 39449, not using umgrnloop0 39354. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 17-Oct-2020.) (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | usgredgne 39451 | An edge of a simple graph always connects two different vertices. Analogue of usgrnloopv 39445 resp. usgrnloop 39447. (Contributed by Alexander van der Vekens, 2-Sep-2017.) (Revised by AV, 17-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Theorem | usgrf1oedg 39452 | The edge function of a simple graph is a 1-1 function onto the set of edges. (Contributed by by AV, 18-Oct-2020.) |
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Theorem | uhgr2edg 39453* | If a vertex is adjacent to two different vertices in a hypergraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.) |
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Theorem | umgr2edg 39454* | If a vertex is adjacent to two different vertices in a multigraph, there are more than one edges starting at this vertex. (Contributed by Alexander van der Vekens, 10-Dec-2017.) (Revised by AV, 11-Feb-2021.) |
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Theorem | usgr2edg 39455* | 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.) (Proof shortened by AV, 11-Feb-2021.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | usgr2edg1 39456* | 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.) |
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Theorem | usgredg3 39457* | 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.) |
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Theorem | usgredg4 39458* | 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.) |
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Theorem | usgredgreu 39459* | 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.) |
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Theorem | usgredg2vtx 39460* | For a vertex incident to an edge there is another vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 5-Dec-2020.) |
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Theorem | uspgredg2vtxeu 39461* | For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple pseudograph. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 6-Dec-2020.) |
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Theorem | usgredg2vtxeu 39462* | For a vertex incident to an edge there is exactly one other vertex incident to the edge in a simple graph. (Contributed by AV, 18-Oct-2020.) (Proof shortened by AV, 6-Dec-2020.) |
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Theorem | usgredg2vtxeuALT 39463* |
Alternate proof of usgredg2vtxeu 39462, using edgiedgb 39377, the general
translation from ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | uspgredg2vlem 39464* | Lemma for uspgredg2v 39465. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.) |
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Theorem | uspgredg2v 39465* | In a simple pseudograph, the mapping of edges having a fixed endpoint to the "other" vertex of the edge (which may be the fixed vertex itself in the case of a loop) is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 6-Dec-2020.) |
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Theorem | usgredg2vlem1 39466* | Lemma 1 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
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Theorem | usgredg2vlem2 39467* | Lemma 2 for usgredg2v 39468. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
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Theorem | usgredg2v 39468* | In a simple graph, the mapping of edges having a fixed endpoint to the other vertex of the edge is a one-to-one function into the set of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 18-Oct-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | usgredgleord 39469* | 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.) |
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Theorem | ushgredgedga 39470* | In a simple hypergraph 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 AV, 11-Dec-2020.) |
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Theorem | usgredgedga 39471* | 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.) (Proof shortened by AV, 11-Dec-2020.) |
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Theorem | ushgredgedgaloop 39472* | In a simple hypergraph there is a 1-1 onto mapping between the indexed edges being loops at a fixed vertex and the set of loops at this vertex. (Contributed by AV, 11-Dec-2020.) |
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Theorem | uspgredgaleord 39473* | In a simple pseudograph 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, 6-Dec-2020.) |
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Theorem | usgredgaleord 39474* | 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.) (Proof shortened by AV, 6-Dec-2020.) |
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Theorem | usgredgaleordALT 39475* | 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.) (Proof modification is discouraged.) (New usage is discouraged.) |
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Theorem | usgr0e 39476 | 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.) (Proof shortened by AV, 25-Nov-2020.) |
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Theorem | usgr0vb 39477 | 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.) |
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Theorem | uhgr0v0e 39478 | The null graph, with no vertices, has no edges. (Contributed by AV, 21-Oct-2020.) |
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Theorem | uhgr0vsize0 39479 | 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.) |
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Theorem | usgr0v 39480 | The null graph, with no vertices, is a simple graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | uhgr0vusgr 39481 | The null graph, with no vertices, represented by a hypergraph, is a simple graph. (Contributed by AV, 5-Dec-2020.) |
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Theorem | usgr0 39482 | The null graph represented by an empty set is a simple graph. (Contributed by AV, 16-Oct-2020.) |
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Theorem | uspgr1e 39483 | A simple pseudograph with one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 21-Mar-2021.) |
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Theorem | usgr1e 39484 |
A simple graph with one edge ( with additional assumption that
![]() ![]() ![]() |
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Theorem | usgr0eop 39485 | 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.) |
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Theorem | uspgr1eop 39486 | A simple pseudograph with (at least) two vertices and one edge. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
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Theorem | uspgr1ewop 39487 | A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
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Theorem | uspgr1v1eop 39488 | A simple pseudograph with (at least) one vertex and one edge (a loop). (Contributed by AV, 5-Dec-2020.) |
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Theorem | usgr1eop 39489 | A simple graph with (at least) two different vertices and one edge. If the two vertices were not different, the edge would be a loop. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 18-Oct-2020.) |
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Theorem | uspgr2v1e2w 39490 | A simple pseudograph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
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Theorem | usgr2v1e2w 39491 | A simple graph with two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021.) |
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Theorem | edg0usgr 39492 |
A class without edges is a simple graph. Since ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | usgr1vr 39493 | A simple graph with one vertex has no edges. (Contributed by AV, 18-Oct-2020.) (Revised by AV, 21-Mar-2021.) |
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Theorem | usgr1v 39494 | A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.) |
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Theorem | usgr1v0edg 39495 | A class with one (or no) vertex is a simple graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 18-Oct-2020.) |
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Theorem | usgrexmpllem 39496 | Lemma for usgrexmpl 39499. (Contributed by AV, 21-Oct-2020.) |
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Theorem | usgrexmplvtx 39497 |
The vertices ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | usgrexmpledg 39498 |
The edges ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | usgrexmpl 39499 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | griedg0prc 39500* | The class of empty graphs (represented as ordered pairs) is a proper class. (Contributed by AV, 27-Dec-2020.) |
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