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

Theoremnbgranself2 21401 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
USGrph Neighbors

Theoremnbgrasym 21402 A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
USGrph Neighbors Neighbors

Theoremnbgracnvfv 21403 Applying the edge function on the converse edge function applied on a pair of a vertex and one of its neighbors is this pair. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
USGrph Neighbors

Theoremnbgraf1olem1 21404* Lemma 1 for nbgraf1o 21410. For each neighbor of a vertex there is exacly one index for the edge between the vertex and its neighbor. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Neighbors                      USGrph

Theoremnbgraf1olem2 21405* Lemma 2 for nbgraf1o 21410. The mapping of neighbors to edge indices is a function. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Neighbors                      USGrph

Theoremnbgraf1olem3 21406* Lemma 3 for nbgraf1o 21410. The restricted iota of an edge is the function value of the converse applied to the edge. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Neighbors                      USGrph

Theoremnbgraf1olem4 21407* Lemma 4 for nbgraf1o 21410. The mapping of neighbors to edge indices applied on a neighbor is the function value of the converse applied on the edge between the vertex and this neighbor. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
Neighbors                      USGrph

Theoremnbgraf1olem5 21408* Lemma 5 for nbgraf1o 21410. The mapping of neighbors to edge indices is a one-to-one onto function. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Neighbors                      USGrph

Theoremnbgraf1o0 21409* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
Neighbors               USGrph

Theoremnbgraf1o 21410* The set of neighbors of a vertex is isomorphic to the set of indices of edges containing the vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
USGrph Neighbors

Theoremnbusgrafi 21411 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
USGrph Neighbors

Theoremedgusgranbfin 21412* The number of neighbors of a vertex in a graph is finite, if and only if the number of edges having this vertex as endpoint is finite. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph Neighbors

Theoremnb3graprlem1 21413 Lemma 1 for nb3grapr 21415. (Contributed by Alexander van der Vekens, 15-Oct-2017.)
USGrph Neighbors

Theoremnb3graprlem2 21414* Lemma 2 for nb3grapr 21415. (Contributed by Alexander van der Vekens, 17-Oct-2017.)
USGrph Neighbors Neighbors

Theoremnb3grapr 21415* 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

Theoremnb3grapr2 21416 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 21417 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 21418* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
ComplUSGrph USGrph

Theoremiscusgra0 21419* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph USGrph

Theoremcusisusgra 21420 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph USGrph

Theoremcusgrarn 21421* 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 21422 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph

Theoremcusgra1v 21423 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
ComplUSGrph

Theoremcusgra2v 21424 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 21425 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 21426 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 21427* 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 21428* Lemma 1 for cusgraexi 21430. (Contributed by Alexander van der Vekens, 12-Jan-2018.)

Theoremcusgraexilem2 21429* Lemma 2 for cusgraexi 21430. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
USGrph

Theoremcusgraexi 21430* 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 21431* 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 21432 Base case of the induction in cusgrasize 21440. 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 21433 Base case of the induction in cusgrasize 21440. 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 21434* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
ComplUSGrph ComplUSGrph

Theoremcusgrasizeindslem1 21435* Lemma 1 for cusgrasizeinds 21438. 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 21436* Lemma 2 for cusgrasizeinds 21438. (Contributed by Alexander van der Vekens, 11-Jan-2018.)

Theoremcusgrasizeindslem3 21437* Lemma 3 for cusgrasizeinds 21438. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
ComplUSGrph

Theoremcusgrasizeinds 21438* Part 1 of induction step in cusgrasize 21440. 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 21439* Induction step in cusgrasize 21440. 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 21440 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 21441* Lemma 1 for cusgrafi 21444. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
ComplUSGrph

Theoremcusgrafilem2 21442* Lemma 2 for cusgrafi 21444. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Theoremcusgrafilem3 21443* Lemma 2 for cusgrafi 21444. (Contributed by Alexander van der Vekens, 13-Jan-2018.)

Theoremcusgrafi 21444 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 21445* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusglem1 21446 Lemma 1 for sizeusglecusg 21448. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusglem2 21447 Lemma 2 for sizeusglecusg 21448. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
USGrph ComplUSGrph

Theoremsizeusglecusg 21448 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 21449 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 21450* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxel 21451* An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtxisvtx 21452 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtx0 21453 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
UnivVertex

Theoremuvtx01vtx 21454* 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 21455 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnm1nbgra 21456 A universal vertex has neighbors in a graph with vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremuvtxnbgravtx 21457* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
USGrph UnivVertex Neighbors

Theoremcusgrauvtxb 21458 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 21459 Extend class notation with Walks (of a graph).
Walks

Syntaxctrail 21460 Extend class notation with Trails (of a graph).
Trails

Syntaxcpath 21461 Extend class notation with Paths (of a graph).
Paths

Syntaxcspath 21462 Extend class notation with Simple Paths (of a graph).
SPaths

Syntaxcwlkon 21463 Extend class notation with Walks between two vertices (within a graph).
WalkOn

Syntaxctrlon 21464 Extend class notation with Trails between two vertices (within a graph).
TrailOn

Syntaxcpthon 21465 Extend class notation with Paths between two vertices (within a graph).
PathOn

Syntaxcspthon 21466 Extend class notation with simple paths between two vertices (within a graph).
SPathOn

Syntaxccrct 21467 Extend class notation with Circuits (of a graph).
Circuits

Syntaxccycl 21468 Extend class notation with Cycles (of a graph).
Cycles

Definitiondf-wlk 21469* 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 21470* 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 21471* 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 21561).

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 21472* 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 21473* 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 21474* 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 21475* 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 21476* 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 21477* 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

Definitiondf-spthon 21478* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn WalkOn SPaths

14.1.5.1  Walks and trails

Theoremwlks 21479* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
Walks Word ..^

Theoremiswlk 21480* 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 21481 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Walks Word

Theoremwlkres 21482* Restrictions of walks are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Walks

Theoremwlkon 21483* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremiswlkon 21484 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 21485 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkoniswlk 21486 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
WalkOn Walks

Theoremwlkbprop 21487 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
Walks

Theoremwlkonwlk 21488 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
Walks WalkOn

Theoremtrls 21489* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Word ..^

Theoremistrl 21490* 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 21491* 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 21492 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
Trails Walks

Theoremtrlon 21493* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
TrailOn WalkOn Trails

Theoremistrlon 21494 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 21495 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 21496 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
TrailOn Trails

Theoremtrlonwlkon 21497 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
TrailOn WalkOn

Theorem0wlk 21498 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 21499 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 21500 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
WalkOn

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