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Type | Label | Description |
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Statement | ||
Theorem | nb3grprlem2 39601* | Lemma 2 for nb3grapr 25237. (Contributed by Alexander van der Vekens, 17-Oct-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nb3grpr 39602* | The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nb3grpr2 39603 | The neighbors of a vertex in a simple graph with three elements are an unordered pair of the other vertices iff all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nb3gr2nb 39604 | If the neighbors of two vertices in a graph with three elements are an unordered pair of the other vertices, the neighbors of all three vertices are an unordered pair of the other vertices. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | uvtxaval 39605* | The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) |
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Theorem | uvtxael 39606* | A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.) |
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Theorem | uvtxaisvtx 39607 | A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtxassvtx 39608 | The set of the universal vertices is a subset of the set of all vertices. (Contributed by AV, 23-Dec-2020.) |
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Theorem | vtxnbuvtx 39609* | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtxanbgr 39610 | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtxanbgrvtx 39611* | A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtxa0 39612 | There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | isuvtxa 39613* | The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtxael1 39614* | A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) |
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Theorem | uvtxa01vtx0 39615 | If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by AV, 30-Oct-2020.) |
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Theorem | uvtxa01vtx 39616 | If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) |
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Theorem | uvtx2vtx1edg 39617* | If a graph has two vertices, and there is an edge between the vertices, then each vertex is universal. (Contributed by AV, 1-Nov-2020.) |
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Theorem | uvtx2vtx1edgb 39618* | If a hypergraph has two vertices, there is an edge between the vertices iff each vertex is universal. (Contributed by AV, 3-Nov-2020.) |
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Theorem | uvtxnbgr 39619 | A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
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Theorem | uvtxnbgrb 39620 | A vertex is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) |
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Theorem | uvtxusgr 39621* | The set of all universal vertices of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.) |
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Theorem | uvtxusgrel 39622* | A universal vertex, i.e. an element of the set of all universal vertices, of a simple graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 31-Oct-2020.) |
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Theorem | uvtxanm1nbgr 39623 |
A universal vertex has ![]() ![]() ![]() ![]() |
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Theorem | nbusgrvtxm1uvtx 39624 | If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.) |
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Theorem | uvtxnbvtxm1 39625 |
A universal vertex has ![]() ![]() ![]() ![]() |
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Theorem | nbupgruvtxres 39626* | The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
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Theorem | uvtxupgrres 39627* | A universal vertex is universal in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
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Theorem | iscplgr 39628* | The property of being a complete graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cplgruvtxb 39629 | An graph is complete iff each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | iscplgrnb 39630* | A graph is complete iff all vertices are neighbors of all vertices. (Contributed by AV, 1-Nov-2020.) |
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Theorem | iscplgredg 39631* | A graph is complete iff all vertices are connected with each other by (at least) one edge. (Contributed by AV, 10-Nov-2020.) |
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Theorem | iscusgr 39632 | The property of being a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cusgrusgr 39633 | A complete simple graph is a simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cusgrcplgr 39634 | A complete simple graph is a complete graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | iscusgrvtx 39635* | A simple graph is complete iff all vertices are uniuversal. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cusgruvtxb 39636 | A simple graph is complete iff the set of vertices is the set of universal vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.) (Revised by AV, 1-Nov-2020.) |
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Theorem | iscusgredg 39637* | A simple graph is complete iff all vertices are connected by an edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cusgredg 39638* | In a complete simple graph, the edges are all the pairs of different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cplgr0 39639 | The null graph (with no vertices and no edges) represented by the empty set is a complete graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cusgr0 39640 | The null graph (with no vertices and no edges) represented by the empty set is a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cplgr0v 39641 | A graph with no vertices (and therefore no edges) is a complete graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cusgr0v 39642 | A graph with no vertices (and therefore no edges) is a complete simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cplgr1v 39643 | A graph with one vertex is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.) (Revised by AV, 1-Nov-2020.) |
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Theorem | cusgr1v 39644 | A graph with one vertex and no edges is a complete simple graph. (Contributed by AV, 1-Nov-2020.) |
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Theorem | cplgr2v 39645 | An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by AV, 3-Nov-2020.) |
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Theorem | cplgr2vpr 39646 | An undirected hypergraph with two (different) vertices is complete iff there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.) (Revised by AV, 3-Nov-2020.) |
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Theorem | nbcplgr 39647 | In a complete graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
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Theorem | cplgr3v 39648 | A pseudograph with three (different) vertices is complete iff there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 5-Nov-2020.) |
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Theorem | cusgr3vnbpr 39649* | The neighbors of a vertex in a simple graph with three elements are unordered pairs of the other vertices if and only if the graph is complete. (Contributed by Alexander van der Vekens, 18-Oct-2017.) (Revised by AV, 5-Nov-2020.) |
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Theorem | cplgrop 39650 | A complete graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.) |
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Theorem | cusgrop 39651 | A complete simple graph represented by an ordered pair. (Contributed by AV, 10-Nov-2020.) |
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Theorem | usgrexi 39652* | An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
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Theorem | cusgrexi 39653* | An arbitrary set regarded as vertices together with the set of pairs of elements of this set regarded as edges is a complete simple graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
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Theorem | cusgrexg 39654* | For each set there is a set of edges so that the set together with these edges is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 5-Nov-2020.) |
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Theorem | cusgrres 39655* | Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) |
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Theorem | cusgrsizeindb0 39656 | Base case of the induction in cusgrasize 25262. The size of a complete simple graph with 0 vertices, actually of every null graph, is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
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Theorem | cusgrsizeindb1 39657 | Base case of the induction in cusgrasize 25262. The size of a (complete) simple graph with 1 vertex is 0=((1-1)*1)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 7-Nov-2020.) |
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Theorem | cusgrsizeindslem 39658* | Lemma for cusgrsizeinds 39659. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
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Theorem | cusgrsizeinds 39659* |
Part 1 of induction step in cusgrsize 39661. The size of a complete
simple graph with ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cusgrsize2inds 39660* |
Induction step in cusgrasize 25262. If the size of the complete graph
with ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cusgrsize 39661 |
The size of a finite complete simple graph with ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | cusgrfilem1 39662* | Lemma 1 for cusgrfi 39665. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
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Theorem | cusgrfilem2 39663* | Lemma 2 for cusgrfi 39665. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
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Theorem | cusgrfilem3 39664* | Lemma 3 for cusgrfi 39665. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
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Theorem | cusgrfi 39665 | If the size of a complete simple graph is finite, then its order is also finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.) |
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Theorem | usgredgsscusgredg 39666 | A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
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Theorem | usgrsscusgr 39667* | A simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
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Theorem | sizusglecusglem1 39668 | Lemma 1 for sizusglecusg 39670. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
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Theorem | sizusglecusglem2 39669 | Lemma 2 for sizusglecusg 39670. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
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Theorem | sizusglecusg 39670 |
The size of a simple graph with ![]() ![]() ![]() |
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Theorem | fusgrmaxsize 39671 |
The maximum size of a finite simple graph with ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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The definition df-vdgr 25678 of the vertex degree VDeg is independent of
the
representation (or even the existence) of a graph. Therefore, it could be used
for the revised definitions of graphs without modification. The way to use
the set of vertices and the edge function separately differs from the way to
use a class | ||
Syntax | cvtxdg 39672 | Extend class notation with the vertex degree function. |
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Definition | df-vtxdg 39673* |
Define the vertex degree function for a graph. To be appropriate for
arbitrary hypergraphs, we have to double-count those edges that contain
![]() ![]() ![]() |
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Theorem | vtxdgfval 39674* | The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.) |
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Theorem | vtxdgval 39675* | The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
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Theorem | vtxdgfival 39676* | The degree of a vertex for graphs of finite size. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 8-Dec-2020.) |
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Theorem | vtxdgf 39677 | The vertex degree function is a function from vertices to extended nonnegative integers. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
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Theorem | vtxdgelxnn0 39678 | The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-Dec-2020.) |
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Theorem | vtxdg0v 39679 | The degree of a vertex in the null graph is zero (or anything else), because there are no vertices. (Contributed by AV, 11-Dec-2020.) |
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Theorem | vtxdg0e 39680 | The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
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Theorem | vtxdgfisnn0 39681 | The degree of a vertex in a graph of finite size is a nonnegative integer. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 11-Dec-2020.) |
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Theorem | vtxdgfisf 39682 | The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
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Theorem | vtxduhgr0e 39683 | The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 39680. (Contributed by AV, 15-Dec-2020.) |
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Theorem | vtxduhgrun 39684 | The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) |
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Theorem | vtxduhgrfiun 39685 | The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) |
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Theorem | vtxdumgrval 39686* | The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.) |
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Theorem | vtxdusgrval 39687* | 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.) |
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Theorem | vtxdushgrfvedglem 39688* | Lemma for vtxdushgrfvedg 39689 and vtxdusgrfvedg 39690. (Contributed by AV, 12-Dec-2020.) TODO-AV: proof can be shortened by using "bj-eleq2w", after it is moved to main.set. |
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Theorem | vtxdushgrfvedg 39689* | The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.) |
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Theorem | vtxdusgrfvedg 39690* | The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.) |
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Theorem | uhgrvd0nedgb 39691* | A vertex in a hypergraph has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) |
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Theorem | vtxduhgr0nedg 39692* | 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.) |
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Theorem | vtxdumgr0nedg 39693* | 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.) |
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Theorem | vtxduhgr0edgnel 39694* | A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020.) |
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Theorem | vtxdusgr0edgnel 39695* | 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.) |
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Theorem | vtxdusgr0edgnelALT 39696* | Alternate proof of vtxdusgr0edgnel 39695, not based on vtxduhgr0edgnel 39694. 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.) |
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Theorem | vtxdgfusgrf 39697 | The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.) |
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Theorem | vtxdgfusgr 39698* | 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.) |
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Theorem | uspgrloopvtx 39699 | The set of vertices in a simple pseudograph with one edge which is a loop (see uspgr1v1eop 39470). (Contributed by AV, 17-Dec-2020.) |
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Theorem | uspgrloopvtxel 39700 | A vertex in a simple pseudograph with one edge which is a loop (see uspgr1v1eop 39470). (Contributed by AV, 17-Dec-2020.) |
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