Home Metamath Proof ExplorerTheorem List (p. 258 of 411) < Previous  Next > Browser slow? Try the Unicode version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26652) Hilbert Space Explorer (26653-28175) Users' Mathboxes (28176-41046)

Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-vdgr 25701* 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 25702* 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 25703* 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 25704* 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 25705 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 25706 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 25707 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 25708 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 25709 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 25710 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 25711 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 25712 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 25713 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 25714* The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph VDeg

Theoremvdusgra0nedg 25715* 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 25716 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
USGrph VDeg

Theoremusgfidegfi 25717* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
USGrph VDeg

Theoremusgfiregdegfi 25718* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
USGrph VDeg

Theoremhashnbgravd 25719 The size of the set of the neighbors of a vertex is the vertex degree of this vertex. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
USGrph Neighbors VDeg

Theoremhashnbgravdg 25720 The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 25719. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph Neighbors VDeg

Theoremnbhashnn0 25721 The number of the neighbors of a vertex in a finite undirected simple graph is a nonnegative integer. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph Neighbors

Theoremnbhashuvtx1 25722 If the number of the neighbors of a vertex in a finite 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.)
USGrph Neighbors Neighbors

Theoremnbhashuvtx 25723 If the number of the neighbors of a vertex in a graph is the number of vertices of the graph minus 1, the vertex is universal. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph Neighbors UnivVertex

Theoremuvtxhashnb 25724 A universal vertex has neighbors in a graph with vertices, a biconditional version of uvtxnm1nbgra 25301. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph UnivVertex Neighbors

Theoremusgravd0nedg 25725* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge, analogous to vdusgra0nedg 25715. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph VDeg

Theoremusgravd00 25726* If every vertex in a simple graph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
USGrph VDeg

Theoremusgrauvtxvdbi 25727 In a finite undirected simple graph with n vertices a vertex is universal if the vertex has degree . (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph UnivVertex VDeg

Theoremvdiscusgra 25728* In a finite complete undirected simple graph with n vertices every vertex has degree . (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph VDeg ComplUSGrph

16.1.7  Regular graphs

16.1.7.1  Definition and basic properties

Syntaxcrgra 25729 Extend class notation to include the class of all regular graphs.
RegGrph

Syntaxcrusgra 25730 Extend class notation to include the class of all regular undirected simple graphs.
RegUSGrph

Definitiondf-rgra 25731* Define the class of k-regular "graphs". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
RegGrph VDeg

Definitiondf-rusgra 25732* Define the class of k-regular undirected simple graphs. (Contributed by Alexander van der Vekens, 6-Jul-2018.)
RegUSGrph USGrph RegGrph

Theoremisrgra 25733* The property of being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
RegGrph VDeg

Theoremisrusgra 25734* The property of being a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.)
RegUSGrph USGrph VDeg

Theoremrgraprop 25735* The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
RegGrph VDeg

Theoremrusgraprop 25736* The properties of a k-regular undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
RegUSGrph USGrph VDeg

Theoremrusgrargra 25737 A k-regular undirected simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
RegUSGrph RegGrph

Theoremrusisusgra 25738 Any k-regular undirected simple graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
RegUSGrph USGrph

Theoremisrusgusrg 25739 A graph is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 3-Jan-2020.)
RegUSGrph USGrph RegGrph

Theoremisrusgusrgcl 25740 A graph represented by a class is a k-regular undirected simple graph iff it is an undirected simple graph and a k-regular graph. (Contributed by AV, 2-Jan-2020.)
RegUSGrph USGrph RegGrph

Theoremisrgrac 25741* The property of being a k-regular graph represented by a class. (Contributed by AV, 3-Jan-2020.)
RegGrph VDeg

Theoremisrusgrac 25742* The property of being a k-regular undirected simple graph represented by a class. (Contributed by AV, 3-Jan-2020.)
RegUSGrph USGrph VDeg

Theorem0egra0rgra 25743 A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.)
RegGrph

Theorem0vgrargra 25744* A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
RegGrph

Theoremcusgraisrusgra 25745 A complete undirected simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.)
ComplUSGrph RegUSGrph

Theorem0eusgraiff0rgra 25746 An undirected simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
USGrph RegGrph

Theoremcusgraiffrusgra 25747 A finite undirected simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGrph allowed for , then the assumption could be removed. Furthermore, if the definition of RegGrph also allowed for , then the theorem would also hold for inifinite graphs. (Contributed by Alexander van der Vekens, 14-Jul-2018.)
USGrph ComplUSGrph RegUSGrph

Theorem0eusgraiff0rgracl 25748 An undirected simple graph represented by a class is 0-regular iff it has no edges. (Contributed by AV, 3-Jan-2020.)
USGrph RegGrph Edges

Theoremrusgraprop2 25749* The properties of a k-regular undirected simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
RegUSGrph USGrph Neighbors

Theoremrusgraprop3 25750* The properties of a k-regular undirected simple graph expressed with edges. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
RegUSGrph USGrph

Theoremrusgraprop4 25751* The properties of a k-regular undirected simple graph expressed with trailing edges of walks (as words). (Contributed by Alexander van der Vekens, 2-Aug-2018.)
RegUSGrph USGrph Word lastS

Theoremrusgrasn 25752 If a k-regular undirected simple graph has only one vertex, then k must be 0. (Contributed by Alexander van der Vekens, 4-Sep-2018.)
RegUSGrph

16.1.7.2  Walks in regular graphs

Theoremrusgranumwwlkl1 25753* In a k-regular graph, the number of walks of length 1 represented as words (thus the number of edges) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
RegUSGrph Word

Theoremrusgranumwlkl1 25754* In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
RegUSGrph WWalksN

Theoremrusgranumwlklem0 25755* Lemma 0 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 23-Aug-2018.)

Theoremrusgranumwlklem1 25756* Lemma 1 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Walks        Walks

Theoremrusgranumwlklem2 25757* Lemma 2 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Walks

Theoremrusgranumwlklem3 25758* Lemma 3 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 21-Jul-2018.)
Walks               Walks

Theoremrusgranumwlklem4 25759* Lemma 4 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 24-Jul-2018.)
Walks               USGrph WWalksN

Theoremrusgranumwlkb0 25760* Induction base 0 for rusgranumwlk 25764. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.)
Walks               USGrph

Theoremrusgranumwlkb1 25761* Induction base 1 for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 28-Jul-2018.)
Walks               RegUSGrph

Theoremrusgra0edg 25762* Special case for graphs without edges: There are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.)
Walks               RegUSGrph

Theoremrusgranumwlks 25763* Induction step for rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
Walks               RegUSGrph

Theoremrusgranumwlk 25764* In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. We denote with the set of walks with length n (in a given undirected simple graph) and with the number of walks with length n starting at the vertex v. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". Because of the k-regularity, the walk can be continued in k different ways at each vertex in the walk, therefore n times. This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
Walks               RegUSGrph

Theoremrusgranumwlkg 25765* In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". This theorem even holds for n=0: then the walk consists only of one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. Closed form of rusgranumwlk 25764. (Contributed by Alexander van der Vekens, 24-Aug-2018.)
RegUSGrph Walks

Theoremrusgranumwwlkg 25766* In a k-regular graph, the number of walks (represented by words) of a fixed length n from a fixed vertex is k to the power of n. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
RegUSGrph WWalksN

Theoremclwlknclwlkdifs 25767 The set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between the set of walks of length n starting with this vertex and the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
WWalksN lastS        WWalksN lastS        WWalksN

Theoremclwlknclwlkdifnum 25768* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
WWalksN lastS        WWalksN lastS        RegUSGrph

16.2  Eulerian paths and the Konigsberg Bridge problem

16.2.1  Eulerian paths

Syntaxceup 25769 Extend class notation with Eulerian paths.
EulPaths

Definitiondf-eupa 25770* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Theoremreleupa 25771 The set EulPaths of all Eulerian paths on is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremiseupa 25772* The property " is an Eulerian path on the graph ". An Eulerian path is defined as bijection from the edges to a set a function into the vertices such that for each , is an edge from to . (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths UMGrph

Theoremeupagra 25773 If an eulerian path exists, then is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Theoremeupai 25774* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupatrl 25775* An Eulerian path is a trail.

Unfortunately, the edge function of an Eulerian path has the domain , whereas the edge functions of all kinds of walks defined here have the domain ..^ (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 25441, fargshiftfv 25442, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

..^        EulPaths Trails

Theoremeupacl 25776 An Eulerian path has length , which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaf1o 25777 The function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupafi 25778 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupapf 25779 The function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaseg 25780 The -th edge in an eulerian path is the edge from to . (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupa0 25781 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupares 25782 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths                                    EulPaths

Theoremeupap1 25783 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths                                    EulPaths

Theoremeupath2lem1 25784 Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem2 25785 Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem3 25786* Lemma for eupath2 25787. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths                             VDeg        VDeg

Theoremeupath2 25787* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths        VDeg

Theoremeupath 25788* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths VDeg

16.2.2  The Konigsberg Bridge problem

Theoremvdeg0i 25789 The base case for the induction for calculating the degree of a vertex. The degree of in the empty graph is . (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremumgrabi 25790* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Theoremvdegp1ai 25791* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                                    ++        VDeg

Theoremvdegp1bi 25792* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             ++        VDeg

Theoremvdegp1ci 25793* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             ++        VDeg

Theoremkonigsberg 25794 The Konigsberg Bridge problem. If is the graph on four vertices , with edges , then vertices each have degree three, and has degree five, so there are four vertices of odd degree and thus by eupath 25788 the graph cannot have an Eulerian path. This is Metamath 100 proof #54. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
EulPaths

16.3  The Friendship Theorem

In this section, the basics for the friendship theorem, which is one from the "100 theorem list" (#83), are provided (subsection "Friendship graphs - basics"), including the definition of friendship graphs df-frgra 25796 as special undirected simple graphs without loops (see frisusgra 25799). In subsection "The friendship theorem for small graphs", the friendship theorem for small graphs (with up to 3 vertices) is proved, see 1to3vfriendship 25815. The general friendship theorem friendship 25929 ( FriendGrph is proven by following the approach of [Huneke] in subsection "Huneke's Proof of the Friendship Theorem". The case (a graph without vertices) must be excluded either from the definition of a friendship graph, or from the theorem. If it is not excluded from the definition, which is the case with df-frgra 25796, a graph without vertices is a friendship graph (see frgra0 25801), but the friendship condition does not hold (because of , see rex0 3737).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 25805, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 25806, a graph with exactly 3 (different) vertices is a friendship graph if and only if it is a complete graph (every two vertices are connected by an edge), see frgra3v 25809, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 25814 (The generalization of this theorem "Every friendship graph (with at least one vertex) is a windmill graph" is a stronger result than the "friendship theorem". This generalization was proven by Mertzios and Unger, see Theorem 1 of [MertziosUnger] p. 152.).

In subsection "Theorems according to Mertzios and Unger", the first steps to prove the friendship theorem following the approach of Mertzios and Unger are made by 2pthfrgrarn2 25817 and n4cyclfrgra 25825 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.).

16.3.1  Friendship graphs - basics

Syntaxcfrgra 25795 Extend class notation with Friendship Graphs.
FriendGrph

Definitiondf-frgra 25796* Define the class of all Friendship Graphs. A graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.)
FriendGrph USGrph

Theoremisfrgra 25797* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
FriendGrph USGrph

Theoremfrisusgrapr 25798* A friendship graph is an undirected simple graph without loops with special properties. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
FriendGrph USGrph

Theoremfrisusgra 25799 A friendship graph is an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
FriendGrph USGrph

Theoremfrgra0v 25800 Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
FriendGrph

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41046
 Copyright terms: Public domain < Previous  Next >