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Type | Label | Description |
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Statement | ||
Theorem | subgreldmiedg 39501 | An element of the domain of the edge function of a subgraph is an element of the domain of the edge function of the supergraph. (Contributed by AV, 20-Nov-2020.) |
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Theorem | subgruhgredgd 39502 | An edge of a subgraph of a hypergraph is a nonempty subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.) |
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Theorem | subumgredg2 39503* | An edge of a subgraph of a multigraph connects exactly two different vertices. (Contributed by AV, 26-Nov-2020.) |
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Theorem | subuhgr 39504 | A subgraph of a hypergraph is a hypergraph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
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Theorem | subupgr 39505 | A subgraph of a pseudograph is a pseudograph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
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Theorem | subumgr 39506 | A subgraph of a multigraph is a multigraph. (Contributed by AV, 26-Nov-2020.) |
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Theorem | subusgr 39507 | A subgraph of a simple graph is a simple graph. (Contributed by AV, 16-Nov-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Theorem | uhgrspansubgrlem 39508 |
Lemma for uhgrspansubgr 39509: The edges of the graph ![]() ![]() |
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Theorem | uhgrspansubgr 39509 |
A spanning subgraph ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | uhgrspan 39510 |
A spanning subgraph ![]() ![]() |
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Theorem | upgrspan 39511 |
A spanning subgraph ![]() ![]() |
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Theorem | umgrspan 39512 |
A spanning subgraph ![]() ![]() |
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Theorem | usgrspan 39513 |
A spanning subgraph ![]() ![]() |
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Theorem | uhgrspanop 39514 | A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) |
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Theorem | upgrspanop 39515 | A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.) |
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Theorem | umgrspanop 39516 | A spanning subgraph of a multigraph represented by an ordered pair is a multigraph. (Contributed by AV, 27-Nov-2020.) |
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Theorem | usgrspanop 39517 | A spanning subgraph of a simple graph represented by an ordered pair is a simple graph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) |
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Theorem | uhgrspan1lem1 39518 | Lemma 1 for uhgrspan1 39521. (Contributed by AV, 19-Nov-2020.) |
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Theorem | uhgrspan1lem2 39519 | Lemma 2 for uhgrspan1 39521. (Contributed by AV, 19-Nov-2020.) |
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Theorem | uhgrspan1lem3 39520 | Lemma 3 for uhgrspan1 39521. (Contributed by AV, 19-Nov-2020.) |
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Theorem | uhgrspan1 39521* |
The induced subgraph ![]() ![]() ![]() |
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Theorem | upgrres1lem1 39522* | Lemma 1 for upgrres1 39526. (Contributed by AV, 7-Nov-2020.) |
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Theorem | umgrres1lem 39523* | Lemma for umgrres1 39527. (Contributed by AV, 27-Nov-2020.) |
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Theorem | upgrres1lem2 39524* | Lemma 2 for upgrres1 39526. (Contributed by AV, 7-Nov-2020.) |
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Theorem | upgrres1lem3 39525* | Lemma 3 for upgrres1 39526. (Contributed by AV, 7-Nov-2020.) |
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Theorem | upgrres1 39526* | A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 39486 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.) |
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Theorem | umgrres1 39527* | A multigraph obtained by removing one vertex and all edges incident with this vertex is a multigraph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 39486 since the domains of the edge functions may not be compatible. (Contributed by AV, 27-Nov-2020.) |
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Theorem | usgrres1 39528* | Restricting a simple graph by removing one vertex results in a simple graph. Remark: This restricted graph is not a subgraph of the original graph in the sense of df-subgr 39486 since the domains of the edge functions may not be compatible. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 10-Jan-2020.) (Revised by AV, 23-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Syntax | cfusgr 39529 | Extend class notation with finite simple graphs. |
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Definition | df-fusgr 39530 | Define the class of all finite undirected simple graphs without loops (called "finite simple graphs" in the following). A finite graph is an undirected simple graph of finite order, i.e. with a finite set of vertices. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
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Theorem | isfusgr 39531 | The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
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Theorem | fusgrvtxfi 39532 | A finite simple graph has a finite set of vertices. (Contributed by AV, 16-Dec-2020.) |
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Theorem | isfusgrf1 39533* | The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
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Theorem | fusgrusgr 39534 | A finite simple graph is a simple graph. (Contributed by AV, 16-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
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Theorem | opfusgr 39535 | A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.) |
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Theorem | usgredgffibi 39536 | The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.) |
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Theorem | fusgredgfi 39537* | In a finite simple graph the number of edges which contain a given vertex is also finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 21-Oct-2020.) |
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Theorem | usgr1v0e 39538 | The size of a (finite) simple graph with 1 vertex is 0. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 22-Oct-2020.) |
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Theorem | usgrfilem 39539* | In a finite simple graph, the number of edges is finite iff the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Nov-2020.) |
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Theorem | fusgrfisbase 39540 | Induction base for fusgrfis 39542. Main work is done in uhgr0v0e 39460. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
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Theorem | fusgrfisstep 39541* | Induction step in fusgrfis 39542: In a finite simple graph, the number of edges is finite if the number of edges not containing one of the vertices is finite. (Contributed by Alexander van der Vekens, 5-Jan-2018.) (Revised by AV, 23-Oct-2020.) |
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Theorem | fusgrfis 39542 | A finite simple graph is of finite size, i.e. has a finite number of edges. (Contributed by Alexander van der Vekens, 6-Jan-2018.) (Revised by AV, 8-Nov-2020.) |
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Syntax | cnbgr 39543 | Extend class notation with neighbors (of a vertex in a graph). |
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Syntax | cuvtxa 39544 | Extend class notation with the universal vertices (in a graph). |
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Syntax | ccplgr 39545 | Extend class notation with (arbitrary) complete graphs. |
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Syntax | ccusgr 39546 | Extend class notation with complete simple graphs. |
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Definition | df-nbgr 39547* |
Define the (open) neighborhood resp. the class of all neighbors of a
vertex (in a graph), see definition in section I.1 of [Bollobas] p. 3 or
definition in section 1.1 of [Diestel]
p. 3. The neighborhood/neighbors
of a vertex are all (other) vertices which are connected with this
vertex by an edge. In contrast to a closed neighborhood, a vertex is
not a neighbor of itself. This definition is applicable even for
arbitrary hypergraphs.
Remark: To distinguish this definition from other definitions for
neighborhoods resp. neighbors (e.g., |
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Theorem | nbgrprc0 39548 | The set of neighbors is empty if the graph or the vertex are proper classes. (Contributed by AV, 26-Oct-2020.) |
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Definition | df-uvtxa 39549* | Define the class of all universal vertices (in graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph) resp. all other vertices are its neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.) |
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Definition | df-cplgr 39550* | Define the class of all complete "graphs". A class/graph is called complete if every pair of distinct vertices is connected by an edge, i.e. each vertex has all other vertices as neighbors. (Contributed by AV, 24-Oct-2020.) |
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Definition | df-cusgr 39551 | Define the class of all complete simple graphs. A simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge, see definition in section 1.1 of [Diestel] p. 3. In contrast, the definition in section I.1 of [Bollobas] p. 3 is based on the size of (finite) complete graphs, see cusgrsize 39661. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 24-Oct-2020.) |
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Theorem | nbgrval 39552* |
The set of neighbors of a vertex ![]() ![]() |
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Theorem | dfnbgr2 39553* | Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) |
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Theorem | dfnbgr3 39554* | Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25209]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) |
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Theorem | nbgrnvtx0 39555 | There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
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Theorem | nbgrel 39556* |
Characterization of a neighbor of a vertex ![]() ![]() |
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Theorem | nbuhgr 39557* |
The set of neighbors of a vertex in a hypergraph. This version of
nbgrval 39552 (with ![]() |
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Theorem | nbupgr 39558* | The set of neighbors of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.) |
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Theorem | nbupgrel 39559 | A neighbor of a vertex in a pseudograph. (Contributed by AV, 5-Nov-2020.) |
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Theorem | nbumgrvtx 39560* | The set of neighbors of a vertex in a multigraph. (Contributed by AV, 27-Nov-2020.) (Proof shortened by AV, 30-Dec-2020.) |
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Theorem | nbumgr 39561* | The set of neighbors of an arbitrary class in a multigraph. (Contributed by AV, 27-Nov-2020.) |
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Theorem | nbusgrvtx 39562* | The set of neighbors of a vertex in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Theorem | nbusgr 39563* | The set of neighbors of an arbitrary class in a simple graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 27-Nov-2020.) |
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Theorem | nbgr2vtx1edg 39564* | If a graph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) |
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Theorem | nbuhgr2vtx1edgblem 39565* | Lemma for nbuhgr2vtx1edgb 39566. This reverse direction of nbgr2vtx1edg 39564 only holds for classes whose edges are subsets of the set of vertices (hypergraphs!) (Contributed by AV, 2-Nov-2020.) |
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Theorem | nbuhgr2vtx1edgb 39566* | If a hypergraph has two vertices, and there is an edge between the vertices, then each vertex is the neighbor of the other vertex. (Contributed by AV, 2-Nov-2020.) |
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Theorem | nbusgreledg 39567 | A class/vertex is a neighbor of another class/vertex in a simple graph iff the vertices are endpoints of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
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Theorem | uhgrnbgr0nb 39568* | A vertex which is not endpoint of an edge has no neighbor in a hypergraph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
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Theorem | nbgr0vtxlem 39569* | Lemma for nbgr0vtx 39570 and nbgr0edg 39571. (Contributed by AV, 15-Nov-2020.) |
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Theorem | nbgr0vtx 39570 | In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) |
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Theorem | nbgr0edg 39571 | In an empty graph (with no edges), every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.) |
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Theorem | nbgr1vtx 39572 | In a graph with one vertex, all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) |
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Theorem | nbgrisvtx 39573 | Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
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Theorem | nbgrssvtx 39574 | The neighbors of a vertex in a graph are a subset of all vertices of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) |
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Theorem | nbgrnself 39575* | A vertex in a graph is not a neighbor of itself. (Contributed by by AV, 3-Nov-2020.) |
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Theorem | usgrnbnself 39576* | A vertex in a simple graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.) |
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Theorem | nbgrnself2 39577 | A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) |
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Theorem | nbgrssovtx 39578 | The neighbors of a vertex are a subset of all vertices except the vertex itself. Stronger version of nbgrssvtx 39574. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) |
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Theorem | nbgrssvwo2 39579 | The neighbors of a vertex are a subset of all vertices except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 3-Nov-2020.) |
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Theorem | usgrnbnself2 39580 | In a simple graph, a class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.) |
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Theorem | usgrnbssovtx 39581 | The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by AV, 3-Nov-2020.) |
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Theorem | usgrnbssvwo2 39582 | The neighbors of a vertex in a simple graph are a subset of all vertices of the graph except the vertex itself and a vertex which is not a neighbor. (Contributed by Alexander van der Vekens, 13-Jul-2018.) (Revised by AV, 27-Oct-2020.) |
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Theorem | nbgrsym 39583 | A vertex in a graph is a neighbor of a second vertex iff the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 27-Oct-2020.) |
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Theorem | nbupgrres 39584* |
The neighborhood of a vertex in a restricted pseudograph (not
necessarily valid for a hypergraph, because ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | usgrnbcnvfv 39585 | Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair in a simple graph. (Contributed by Alexander van der Vekens, 18-Dec-2017.) (Revised by AV, 27-Oct-2020.) |
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Theorem | nbusgredgeu 39586* | For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.) |
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Theorem | edgnbusgreu 39587* | For each edge incident to a vertex there is exactly one neighbor of the vertex also incident to this edge in a simple graph. (Contributed by AV, 28-Oct-2020.) |
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Theorem | nbusgredgeu0 39588* | For each neighbor of a vertex there is exactly one edge between the vertex and its neighbor in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 27-Oct-2020.) |
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Theorem | nbusgrf1o0 39589* | The mapping of neighbors of a vertex to edges incident to the vertex is a bijection ( 1-1 onto function) in a simple graph. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nbusgrf1o1 39590* | The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nbusgrf1o 39591* | The set of neighbors of a vertex is isomorphic to the set of edges containing the vertex in a simple graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nbedgusgr 39592* | The number of neighbors of a vertex is the number of edges at the vertex in a simple graph. (Contributed by AV, 27-Dec-2020.) |
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Theorem | edgusgrnbfin 39593* | The number of neighbors of a vertex in a simple graph is finite iff the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nbusgrfi 39594 | The class of neighbors of a vertex in a simple graph with a finite number of edges is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Revised by AV, 28-Oct-2020.) |
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Theorem | nbfiusgrfi 39595 | The class of neighbors of a vertex in a finite simple graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.) (Revised by AV, 28-Oct-2020.) |
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Theorem | hashnbusgrnn0 39596 | The number of neighbors of a vertex in a finite simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 15-Dec-2020.) |
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Theorem | nbfusgrlevtxm1 39597 | The number of neighbors of a vertex is at most the number of vertices of the graph minus 1 in a finite simple graph. (Contributed by AV, 16-Dec-2020.) |
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Theorem | nbfusgrlevtxm2 39598 | If there is a vertex which is not a neighbor of another vertex, the number of neighbors of the other vertex is at most the number of vertices of the graph minus 2 in a finite simple graph. (Contributed by AV, 16-Dec-2020.) |
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Theorem | nbusgrvtxm1 39599 | If the number of neighbors of a vertex in a finite simple graph is the number of vertices of the graph minus 1, each vertex except the first mentioned vertex is a neighbor of this vertex. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 16-Dec-2020.) |
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Theorem | nb3grprlem1 39600 | Lemma 1 for nb3grapr 25237. (Contributed by Alexander van der Vekens, 15-Oct-2017.) (Revised by AV, 28-Oct-2020.) |
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