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Theorem List for Metamath Proof Explorer - 21401-21500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnb3grapr2 21401 The neighbors of a vertex in a graph with three elements are an unordered pair of the other vertices if and only if all vertices are connected with each other. (Contributed by Alexander van der Vekens, 18-Oct-2017.)
USGrph Neighbors Neighbors Neighbors

Theoremnb3gra2nb 21402 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.)
USGrph Neighbors Neighbors Neighbors Neighbors Neighbors

14.1.4.2  Complete graphs

Theoremiscusgra 21403* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
ComplUSGrph USGrph

Theoremiscusgra0 21404* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph USGrph

Theoremcusisusgra 21405 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph USGrph

Theoremcusgrarn 21406* In a complete simple graph, the range of the edge function consists of all the pairs with different vertices. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
ComplUSGrph

Theoremcusgra0v 21407 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph

Theoremcusgra1v 21408 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph

Theoremcusgra2v 21409 A graph with two (different) vertices is complete if and only if 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.)
USGrph ComplUSGrph

Theoremnbcusgra 21410 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
ComplUSGrph Neighbors

Theoremcusgra3v 21411 A graph with three (different) vertices is complete if and only if there is an edge between each of these three vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
USGrph ComplUSGrph

Theoremcusgra3vnbpr 21412* The neighbors of a vertex in a 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.)
USGrph ComplUSGrph Neighbors

Theoremcusgraexilem1 21413* Lemma 1 for cusgraexi 21415. (Contributed by Alexander van der Vekens, 12-Jan-2018.)

Theoremcusgraexilem2 21414* Lemma 2 for cusgraexi 21415. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
USGrph

Theoremcusgraexi 21415* For each set the identity function restricted to the set of pairs of elements from the given set is an edge function, so that the given set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
ComplUSGrph

Theoremcusgraexg 21416* For each set there is an edge function so that the set together with this edge function is a complete graph. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
ComplUSGrph

Theoremcusgrasizeindb0 21417 Base case of the induction in cusgrasize 21425. The size of a complete simple graph with 0 vertices is 0=((0-1)*0)/2. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
ComplUSGrph

Theoremcusgrasizeindb1 21418 Base case of the induction in cusgrasize 21425. 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.)
ComplUSGrph

Theoremcusgrares 21419* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
ComplUSGrph ComplUSGrph

Theoremcusgrasizeindslem1 21420* Lemma 1 for cusgrasizeinds 21423. The domain of the edge function is the union of the arguments/indices of all edges containing a specific vertex and the arguments/indices of all edges not containing this vertex. (Contributed by Alexander van der Vekens, 4-Jan-2018.)

Theoremcusgrasizeindslem2 21421* Lemma 2 for cusgrasizeinds 21423. (Contributed by Alexander van der Vekens, 11-Jan-2018.)

Theoremcusgrasizeindslem3 21422* Lemma 3 for cusgrasizeinds 21423. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
ComplUSGrph

Theoremcusgrasizeinds 21423* Part 1 of induction step in cusgrasize 21425. The size of a complete simple graph with vertices is plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
ComplUSGrph

Theoremcusgrasize2inds 21424* Induction step in cusgrasize 21425. If the size of the complete graph with vertices reduced by one vertex is " choose 2", the size of the complete graph with vertices is " choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.)
ComplUSGrph

Theoremcusgrasize 21425 The size of a finite complete simple graph with vertices ( ) is (" choose 2") resp. . (Contributed by Alexander van der Vekens, 11-Jan-2018.)
ComplUSGrph

Theoremcusgrafilem1 21426* Lemma 1 for cusgrafi 21429. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
ComplUSGrph

Theoremcusgrafilem2 21427* Lemma 2 for cusgrafi 21429. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Theoremcusgrafilem3 21428* Lemma 2 for cusgrafi 21429. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Theoremcusgrafi 21429 If the size of a complete simple graph is finite, then also its order is finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
ComplUSGrph

Theoremusgrasscusgra 21430* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusglem1 21431 Lemma 1 for sizeusglecusg 21433. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusglem2 21432 Lemma 2 for sizeusglecusg 21433. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusg 21433 The size of an undirected simple graph with vertices is at most the size of a complete simple graph with vertices ( may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
USGrph ComplUSGrph

Theoremusgramaxsize 21434 The maximum size of an undirected simple graph with vertices ( ) is . (Contributed by Alexander van der Vekens, 13-Jan-2018.)
USGrph

14.1.4.3  Universal vertices

Theoremisuvtx 21435* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxel 21436* An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxisvtx 21437 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtx0 21438 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtx01vtx 21439* If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated UnivVertex , but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxnbgra 21440 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnm1nbgra 21441 A universal vertex has neighbors in a graph with vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnbgravtx 21442* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremcusgrauvtxb 21443 An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.)
USGrph ComplUSGrph UnivVertex

14.1.5  Walks, paths and cycles

Syntaxcwalk 21444 Extend class notation with Walks (of a graph).
Walks

Syntaxctrail 21445 Extend class notation with Trails (of a graph).
Trails

Syntaxcpath 21446 Extend class notation with Paths (of a graph).
Paths

Syntaxcspath 21447 Extend class notation with Simple Paths (of a graph).
SPaths

Syntaxcwlkon 21448 Extend class notation with Walks between two vertices (within a graph).
WalkOn

Syntaxctrlon 21449 Extend class notation with Trails between two vertices (within a graph).
TrailOn

Syntaxcpthon 21450 Extend class notation with Paths between two vertices (within a graph).
PathOn

Syntaxccrct 21451 Extend class notation with Circuits (of a graph).
Circuits

Syntaxccycl 21452 Extend class notation with Cycles (of a graph).
Cycles

Definitiondf-wlk 21453* Define the set of all Walks (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Walks Word ..^

Definitiondf-trail 21454* Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Trails Walks

Definitiondf-pth 21455* Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 21528).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Paths Trails ..^ ..^

Definitiondf-spth 21456* Define the set of all Simple Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) (Contributed by Alexander van der Vekens, 20-Oct-2017.)

SPaths Trails

Definitiondf-crct 21457* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Circuits Trails

Definitiondf-cycl 21458* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Cycles Paths

Definitiondf-wlkon 21459* Define the collection of walks with particular endpoints (in an un- directed graph). This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
WalkOn Walks

Definitiondf-trlon 21460* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
TrailOn WalkOn Trails

Definitiondf-pthon 21461* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
PathOn WalkOn Paths

14.1.5.1  Walks and trails

Theoremwlks 21462* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
Walks Word ..^

Theoremiswlk 21463* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Walks Word ..^

Theorem2mwlk 21464 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Walks Word

Theoremwlkres 21465* Restrictions of walks are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Walks

Theoremwlkon 21466* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremiswlkon 21467 Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
WalkOn Walks

Theoremwlkonprop 21468 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkoniswlk 21469 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkbprop 21470 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Walks

Theoremwlkonwlk 21471 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
Walks WalkOn

Theoremtrls 21472* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Word ..^

Theoremistrl 21473* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Word ..^

Theoremistrl2 21474* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails ..^ ..^

Theoremtrliswlk 21475 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Walks

Theoremtrlon 21476* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
TrailOn WalkOn Trails

Theoremistrlon 21477 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
TrailOn WalkOn Trails

Theoremtrlonprop 21478 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
TrailOn WalkOn Trails

Theoremtrlonistrl 21479 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
TrailOn Trails

Theoremtrlonwlkon 21480 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
TrailOn WalkOn

Theorem0wlk 21481 A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Walks

Theorem0trl 21482 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Trails

Theorem0wlkon 21483 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
WalkOn

Theorem0trlon 21484 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
TrailOn

Theoremwlkntrllem1 21485 Lemma 1 for wlkntrl 21490: F is a word over , the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
Word

Theoremwlkntrllem2 21486 Lemma 2 for wlkntrl 21490: The cardinality of F is 2. (Contributed by Alexander van der Vekens, 22-Oct-2017.)

Theoremwlkntrllem3 21487 Lemma 3 for wlkntrl 21490: P is a function on into . (Contributed by Alexander van der Vekens, 22-Oct-2017.)

Theoremwlkntrllem4 21488* Lemma 4 for wlkntrl 21490: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
..^

Theoremwlkntrllem5 21489* Lemma 5 for wlkntrl 21490: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)

Theoremwlkntrl 21490* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that is a simple graph (without loops) only if . (Contributed by Alexander van der Vekens, 22-Oct-2017.)
Walks Trails

Theoremusgrnloop 21491* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
USGrph Walks ..^

14.1.5.2  Paths and simple paths

Theorempths 21492* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Paths Trails ..^ ..^

Theoremspths 21493* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Trails

Theoremispth 21494 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Paths Trails ..^ ..^

Theoremisspth 21495 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Trails

Theorem0pth 21496 A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Paths

Theorem0spth 21497 A pair of an empty set (of edges) and a second set (of vertices) is a simple path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
SPaths

Theorempthistrl 21498 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
Paths Trails

Theoremspthispth 21499 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
SPaths Paths

Theorempthdepisspth 21500 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Paths SPaths

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