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Theorem List for Metamath Proof Explorer - 21501-21600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempthon 21501* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
PathOn WalkOn Paths

Theoremispthon 21502 Properties of a pair of functions to be a path between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.)
PathOn WalkOn Paths

Theorempthonprop 21503 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
PathOn WalkOn Paths

Theorempthonispth 21504 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
PathOn Paths

Theorem0pthon 21505 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem0pthon1 21506 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem0pthonv 21507* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theoremconstr1trl 21508 Construction of a trail from one given edge in a graph. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
Trails

Theorem1pthonlem1 21509 Lemma 1 for 1pthon 21511. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theorem1pthonlem2 21510 Lemma 2 for 1pthon 21511. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theorem1pthon 21511 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
PathOn

Theorem1pthoncl 21512 A path of length 1 from one vertex to another vertex. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
PathOn

Theorem1pthon2v 21513* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
PathOn

Theorem2trllem1 21514 Lemma 1 for constr2trl 21518. (Contributed by Alexander van der Vekens, 4-Dec-2017.)

Theorem2trllem2 21515 Lemma 2 for constr2trl 21518. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
..^

Theorem2trllem3 21516 Lemma 3 for constr2trl 21518. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
..^

Theorem2trllem4 21517 Lemma 4 for constr2trl 21518. (Contributed by Alexander van der Vekens, 5-Dec-2017.)

Theoremconstr2trl 21518 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
Trails

Theorem2pthonlem1 21519 Lemma 1 for 2pthon 21522. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
..^

Theorem2pthonlem2 21520 Lemma 2 for 2pthon 21522. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
..^

Theoremconstr2pth 21521 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
Paths

Theorem2pthon 21522 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
PathOn

Theorem2pthoncl 21523 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
PathOn

Theorem2pthon3v 21524* For a vertex adjacent to two other vertices there is a path of length 2 between these other vertices. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
PathOn

Theoremredwlklem 21525 Lemma for redwlk 21526. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Walks ..^

Theoremredwlk 21526 A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Walks ..^ Walks ..^

Theoremwlkdvspthlem 21527* Lemma for wlkdvspth 21528. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
Word ..^

Theoremwlkdvspth 21528 A walk consisting of different vertices is a simple path. (Contributed by Alexander van der Vekens, 27-Oct-2017.)
Walks SPaths

14.1.5.3  Circuits and cycles

Theoremcrcts 21529* The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Circuits Trails

Theoremcycls 21530* The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles Paths

Theoremiscrct 21531 Properties of a pair of functions to be a circuit (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Circuits Trails

Theoremiscycl 21532 Properties of a pair of functions to be a cycle (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles Paths

Theorem0crct 21533 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Circuits

Theorem0cycl 21534 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles

Theoremcrctistrl 21535 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Circuits Trails

Theoremcyclispth 21536 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles Paths

Theoremcycliscrct 21537 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles Circuits

Theoremcyclnspth 21538 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
Cycles SPaths

Theoremcycliswlk 21539 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
Cycles Walks

Theoremcyclispthon 21540 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
Cycles PathOn

Theoremfargshiftlem 21541 If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
..^

Theoremfargshiftfv 21542* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
..^        ..^

Theoremfargshiftf 21543* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
..^        ..^

Theoremfargshiftf1 21544* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
..^        ..^

Theoremfargshiftfo 21545* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
..^        ..^

Theoremfargshiftfva 21546* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
..^        ..^

Theoremusgrcyclnl1 21547 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
USGrph Cycles

Theoremusgrcyclnl2 21548 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
USGrph Cycles

Theorem3cycl3dv 21549 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
USGrph

Theoremnvnencycllem 21550 Lemma for 3v3e3cycl1 21551 and 4cycl4v4e 21573. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Word

Theorem3v3e3cycl1 21551* If there is a cycle of length 3 in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Cycles

Theoremconstr3lem1 21552 Lemma for constr3trl 21566 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)

Theoremconstr3lem2 21553 Lemma for constr3trl 21566 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.)

Theoremconstr3lem4 21554 Lemma for constr3trl 21566 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)

Theoremconstr3lem5 21555 Lemma for constr3trl 21566 etc. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Theoremconstr3lem6 21556 Lemma for constr3pthlem3 21564. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremconstr3trllem1 21557 Lemma for constr3trl 21566. (Contributed by Alexander van der Vekens, 10-Nov-2017.)
USGrph Word

Theoremconstr3trllem2 21558 Lemma for constr3trl 21566. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
USGrph

Theoremconstr3trllem3 21559 Lemma for constr3trl 21566. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremconstr3trllem4 21560 Lemma for constr3trl 21566. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremconstr3trllem5 21561* Lemma for constr3trl 21566. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
USGrph ..^

Theoremconstr3pthlem1 21562 Lemma for constr3pth 21567. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
..^

Theoremconstr3pthlem2 21563 Lemma for constr3pth 21567. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
..^

Theoremconstr3pthlem3 21564 Lemma for constr3pth 21567. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
..^

Theoremconstr3cycllem1 21565 Lemma for constr3cycl 21568. (Contributed by Alexander van der Vekens, 11-Nov-2017.)

Theoremconstr3trl 21566 Construction of a trail from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
USGrph Trails

Theoremconstr3pth 21567 Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
USGrph Paths

Theoremconstr3cycl 21568 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
USGrph Cycles

Theoremconstr3cyclp 21569 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
USGrph Cycles

Theoremconstr3cyclpe 21570* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
USGrph Cycles

Theorem3v3e3cycl2 21571* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
USGrph Cycles

Theorem3v3e3cycl 21572* If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
USGrph Cycles

Theorem4cycl4v4e 21573* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
Cycles

Theorem4cycl4dv 21574 In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
USGrph Word

Theorem4cycl4dv4e 21575* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
USGrph Cycles

14.1.5.4  Connected graphs

Syntaxcconngra 21576 Extend class notation with connected graphs.
ConnGrph

Definitiondf-conngra 21577* Define the class of all connected graphs. A graph (or, more generally, any pair representing a structure consisting of "vertices" and "edges") is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 21507 and dfconngra1 21578. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph PathOn

Theoremdfconngra1 21578* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
ConnGrph PathOn

Theoremisconngra 21579* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph PathOn

Theoremisconngra1 21580* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph PathOn

Theorem0conngra 21581 A class/graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph

Theorem1conngra 21582 A class/graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
ConnGrph

Theoremcusconngra 21583 A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
ComplUSGrph ConnGrph

14.1.6  Vertex Degree

Syntaxcvdg 21584 Extend class notation with the vertex degree function.
VDeg

Definitiondf-vdgr 21585* Define the vertex degree function (for an undirected multigraph). To be appropriate for multigraphs, we have to double-count those edges that contain "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgrfval 21586* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgrval 21587* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgrfival 21588* The value of the vertex degree function (for graphs of finite size). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
VDeg

Theoremvdgrf 21589 The vertex degree function is a function from vertices to nonnegative integers or plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgrfif 21590 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.)
VDeg

Theoremvdgr0 21591 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.)
VDeg

Theoremvdgrun 21592 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.)
UMGrph        UMGrph               VDeg VDeg VDeg

Theoremvdgrfiun 21593 The degree of a vertex in the union of two graphs (of finite size) on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
UMGrph        UMGrph               VDeg VDeg VDeg

Theoremvdgr1d 21594 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1b 21595 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1c 21596 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1a 21597 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdusgraval 21598* The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph VDeg

Theoremvdusgra0nedg 21599* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
USGrph VDeg

Theoremvdgrnn0pnf 21600 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
USGrph VDeg

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