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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclwlknclwlkdifnum 25701* 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.)
 |-  A  =  { w  e.  ( ( V WWalksN  E ) `
  N )  |  ( ( w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X ) }   &    |-  B  =  { w  e.  (
 ( V WWalksN  E ) `  N )  |  ( ( lastS  `  w )  =  ( w `  0
 )  /\  ( w `  0 )  =  X ) }   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  /\  ( X  e.  V  /\  N  e.  NN )
 )  ->  ( # `  A )  =  ( ( K ^ N )  -  ( # `  B ) ) )
 
16.2  Eulerian paths and the Konigsberg Bridge problem
 
16.2.1  Eulerian paths
 
Syntaxceup 25702 Extend class notation with Eulerian paths.
 class EulPaths
 
Definitiondf-eupa 25703* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- EulPaths  =  ( v  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( v UMGrph  e  /\  E. n  e.  NN0  (
 f : ( 1
 ... n ) -1-1-onto-> dom  e  /\  p : ( 0
 ... n ) --> v  /\  A. k  e.  ( 1
 ... n ) ( e `  ( f `
  k ) )  =  { ( p `
  ( k  -  1 ) ) ,  ( p `  k
 ) } ) ) } )
 
Theoremreleupa 25704 The set  ( V EulPaths  E ) of all Eulerian paths on  <. V ,  E >. is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- 
 Rel  ( V EulPaths  E )
 
Theoremiseupa 25705* The property " <. F ,  P >. is an Eulerian path on the graph  <. V ,  E >.". An Eulerian path is defined as bijection  F from the edges to a set  1 ... N a function  P :
( 0 ... N
) --> V into the vertices such that for each 
1  <_  k  <_  N,  F ( k ) is an edge from  P ( k  -  1 ) to  P
( k ). (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.)
 |-  ( dom  E  =  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1
 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1
 ... n ) ( E `  ( F `
  k ) )  =  { ( P `
  ( k  -  1 ) ) ,  ( P `  k
 ) } ) ) ) )
 
Theoremeupagra 25706 If an eulerian path exists, then 
<. V ,  E >. is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  V UMGrph  E )
 
Theoremeupai 25707* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( ( ( # `  F )  e.  NN0  /\  F : ( 1
 ... ( # `  F ) ) -1-1-onto-> A  /\  P :
 ( 0 ... ( # `
  F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  ( k  -  1
 ) ) ,  ( P `  k ) }
 ) )
 
Theoremeupatrl 25708* An Eulerian path is a trail.

Unfortunately, the edge function  F of an Eulerian path has the domain  ( 1 ... ( # `  F
) ), whereas the edge functions of all kinds of walks defined here have the domain  ( 0..^ ( # `  F
) ) (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 25374, fargshiftfv 25375, 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.)

 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( F ( V EulPaths  E ) P  ->  G ( V Trails  E ) P )
 
Theoremeupacl 25709 An Eulerian path has length 
# ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  ( # `  F )  e. 
 NN0 )
 
Theoremeupaf1o 25710 The  F 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.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  F : ( 1 ... ( # `  F ) ) -1-1-onto-> A )
 
Theoremeupafi 25711 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  A  e.  Fin )
 
Theoremeupapf 25712 The  P function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  F ) ) --> V )
 
Theoremeupaseg 25713 The  N-th edge in an eulerian path is the edge from  P ( N  - 
1 ) to  P ( N ). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  N  e.  ( 1 ... ( # `  F ) ) )  ->  ( E `  ( F `
  N ) )  =  { ( P `
  ( N  -  1 ) ) ,  ( P `  N ) } )
 
Theoremeupa0 25714 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( V  e.  W  /\  A  e.  V )  ->  (/) ( V EulPaths  (/) ) { <. 0 ,  A >. } )
 
Theoremeupares 25715 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.)
 |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  ( 0 ... ( # `  G ) ) )   &    |-  F  =  ( E  |`  ( G
 " ( 1 ...
 N ) ) )   &    |-  H  =  ( G  |`  ( 1 ... N ) )   &    |-  Q  =  ( P  |`  ( 0 ... N ) )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupap1 25716 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  =  ( # `  G ) )   &    |-  F  =  ( E  u.  { <. B ,  { ( P `
  N ) ,  C } >. } )   &    |-  H  =  ( G  u.  { <. ( N  +  1 ) ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupath2lem1 25717 Lemma for eupath2 25720. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupath2lem2 25718 Lemma for eupath2 25720. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  B  e.  _V   =>    |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
 
Theoremeupath2lem3 25719* Lemma for eupath2 25720. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( N  +  1 ) 
 <_  ( # `  F ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  ( E  |`  ( F "
 ( 1 ... N ) ) ) ) `
  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   =>    |-  ( ph  ->  ( -.  2  ||  ( ( V VDeg  ( E  |`  ( F
 " ( 1 ... ( N  +  1 ) ) ) ) ) `  U )  <->  U  e.  if (
 ( P `  0
 )  =  ( P `
  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupath2 25720* 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.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( # `
  F ) ) ,  (/) ,  { ( P `  0 ) ,  ( P `  ( # `
  F ) ) } ) )
 
Theoremeupath 25721* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( V EulPaths  E )  =/=  (/)  ->  ( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }
 )  e.  { 0 ,  2 } )
 
16.2.2  The Konigsberg Bridge problem
 
Theoremvdeg0i 25722 The base case for the induction for calculating the degree of a vertex. The degree of  U in the empty graph is  0. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  V  e.  _V   &    |-  U  e.  V   =>    |-  ( ( V VDeg  (/) ) `  U )  =  0
 
Theoremumgrabi 25723* 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.)
 |-  V  e.  _V   &    |-  X  e.  V   &    |-  Y  e.  V   =>    |-  ( ph  ->  { X ,  Y }  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremvdegp1ai 25724* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  Y } to the edge set, where  X  =/=  U  =/= 
Y, yields degree  P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  ( T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  Y  e.  V   &    |-  Y  =/=  U   &    |-  F  =  ( E ++  <" { X ,  Y } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  P
 
Theoremvdegp1bi 25725* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { U ,  X } to the edge set, where 
X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  ( T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E ++  <" { U ,  X } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremvdegp1ci 25726* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  U } to the edge set, where  X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  ( T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E ++  <" { X ,  U } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremkonigsberg 25727 The Konigsberg Bridge problem. If  <. V ,  E >. is the graph on four vertices  0 ,  1 ,  2 ,  3, with edges  { 0 ,  1 } ,  { 0 ,  2 } ,  { 0 ,  3 } ,  {
1 ,  2 } ,  { 1 ,  2 } ,  {
2 ,  3 } ,  { 2 ,  3 }, then vertices  0 ,  1 ,  3 each have degree three, and  2 has degree five, so there are four vertices of odd degree and thus by eupath 25721 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.)
 |-  V  =  ( 0
 ... 3 )   &    |-  E  =  <" { 0 ,  1 }  {
 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  { 1 ,  2 }  {
 2 ,  3 }  { 2 ,  3 } ">   =>    |-  ( V EulPaths  E )  =  (/)
 
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 25729 as special undirected simple graphs without loops (see frisusgra 25732). In subsection "The friendship theorem for small graphs", the friendship theorem for small graphs (with up to 3 vertices) is proved, see 1to3vfriendship 25748. The general friendship theorem friendship 25862 ( |-  ( ( V FriendGrph  E  /\  V  =/=  (/)  /\  V  e. 
Fin )  ->  E. v  e.  V A. w  e.  ( V  \  {
v } ) { v ,  w }  e.  ran  E ) is proven by following the approach of [Huneke] in subsection "Huneke's Proof of the Friendship Theorem". The case  V  =  (/) (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 25729, a graph without vertices is a friendship graph (see frgra0 25734), but the friendship condition  E. v  e.  V A. w  e.  ( V  \  {
v } ) { v ,  w }  e.  ran  E does not hold (because of  -.  E. x  e.  (/) ph, see rex0 3748).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 25738, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 25739, 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 25742, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 25747 (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 25750 and n4cyclfrgra 25758 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.).

 
16.3.1  Friendship graphs - basics
 
Syntaxcfrgra 25728 Extend class notation with Friendship Graphs.
 class FriendGrph
 
Definitiondf-frgra 25729* 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  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v  A. l  e.  (
 v  \  { k } ) E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) }
 
Theoremisfrgra 25730* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V FriendGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V 
 \  { k }
 ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E )
 ) )
 
Theoremfrisusgrapr 25731* A friendship graph is an undirected simple graph without loops with special properties. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( V FriendGrph  E  ->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V  \  {
 k } ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E ) )
 
Theoremfrisusgra 25732 A friendship graph is an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( V FriendGrph  E  ->  V USGrph  E )
 
Theoremfrgra0v 25733 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  E  <->  E  =  (/) )
 
Theoremfrgra0 25734 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (/) FriendGrph  (/)
 
Theoremfrgraunss 25735* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran 
 E ) )
 
Theoremfrgraun 25736* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
 
Theoremfrisusgranb 25737* In a friendship graph, the neighborhoods of two different vertices have exactly one vertex in common. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. k  e.  V  A. l  e.  ( V  \  { k } ) E. x  e.  V  ( ( <. V ,  E >. Neighbors  k )  i^i  ( <. V ,  E >. Neighbors  l
 ) )  =  { x } )
 
16.3.2  The friendship theorem for small graphs
 
Theoremfrgra1v 25738 Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( ( V  e.  X  /\  { V } USGrph  E )  ->  { V } FriendGrph  E )
 
Theoremfrgra2v 25739 Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.) (Proof shortened by Alexander van der Vekens, 13-Sep-2018.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  ->  -.  { A ,  B } FriendGrph  E )
 
Theoremfrgra3vlem1 25740* Lemma 1 for frgra3v 25742. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 /\  { A ,  B ,  C } USGrph  E )  ->  A. x A. y
 ( ( ( x  e.  { A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran 
 E )  /\  (
 y  e.  { A ,  B ,  C }  /\  { { y ,  A } ,  {
 y ,  B } }  C_  ran  E )
 )  ->  x  =  y ) )
 
Theoremfrgra3vlem2 25741* Lemma 2 for frgra3v 25742. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( E! x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran 
 E 
 <->  ( { C ,  A }  e.  ran  E 
 /\  { C ,  B }  e.  ran  E ) ) ) )
 
Theoremfrgra3v 25742 Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  B ,  C } FriendGrph  E  <->  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) ) )
 
Theorem1vwmgra 25743* Every graph with one vertex is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  ( ( A  e.  X  /\  V  =  { A } )  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) )
 
Theorem3vfriswmgralem 25744* Lemma for 3vfriswmgra 25745. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
 
Theorem3vfriswmgra 25745* Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C )  /\  V  =  { A ,  B ,  C } )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to2vfriswmgra 25746* Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B } ) )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to3vfriswmgra 25747* Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to3vfriendship 25748* The friendship theorem for small graphs: In every friendship graph with one, two or three vertices, there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  ( ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  ->  ( V FriendGrph  E  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E ) )
 
16.3.3  Theorems according to Mertzios and Unger
 
Theorem2pthfrgrarn 25749* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
 
Theorem2pthfrgrarn2 25750* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  { a } ) E. b  e.  V  ( ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E ) 
 /\  ( a  =/=  b  /\  b  =/=  c ) ) )
 
Theorem2pthfrgra 25751* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1(b) of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. b  e.  ( V  \  { a } ) E. f E. p ( f ( a ( V PathOn  E ) b ) p  /\  ( # `  f )  =  2 ) )
 
Theorem3cyclfrgrarn1 25752* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V ) 
 /\  A  =/=  C )  ->  E. b  e.  V  E. c  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  {
 c ,  A }  e.  ran  E ) )
 
Theorem3cyclfrgrarn 25753* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
 
Theorem3cyclfrgrarn2 25754* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( b  =/=  c  /\  ( {
 a ,  b }  e.  ran  E  /\  {
 b ,  c }  e.  ran  E  /\  {
 c ,  a }  e.  ran  E ) ) )
 
Theorem3cyclfrgra 25755* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3  /\  ( p `  0 )  =  v ) )
 
Theorem4cycl2v2nb 25756 In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) 
 ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran 
 E ) )
 
Theorem4cycl2vnunb 25757* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) ) 
 ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E )
 
Theoremn4cyclfrgra 25758 There is no 4-cycle in a friendship graph, see Proposition 1(a) of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  ( ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  4 )
 
Theorem4cyclusnfrgra 25759 A graph with a 4-cycle is not a friendhip graph. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) ) 
 ->  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) 
 ->  -.  V FriendGrph  E ) )
 
Theoremfrgranbnb 25760 If two neighbors of a specific vertex have a common neighbor in a friendship graph, then this common neighbor must be the specific vertex. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ph  ->  X  e.  V )   &    |-  D  =  (
 <. V ,  E >. Neighbors  X )   &    |-  ( ph  ->  V FriendGrph  E )   =>    |-  ( ( ph  /\  ( U  e.  D  /\  W  e.  D )  /\  U  =/=  W ) 
 ->  ( ( { U ,  A }  e.  ran  E 
 /\  { W ,  A }  e.  ran  E ) 
 ->  A  =  X ) )
 
Theoremfrconngra 25761 A friendship graph is connected, see remark 1 in [MertziosUnger] p. 153 (after Proposition 1): "An arbitrary friendship graph has to be connected, ... ". (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( V FriendGrph  E  ->  V ConnGrph  E )
 
Theoremvdfrgra0 25762 A vertex in a friendship graph has degree 0 if the graph consists of only one vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  N  e.  V  /\  ( # `  V )  =  1 )  ->  ( ( V VDeg  E ) `  N )  =  0 )
 
Theoremvdn0frgrav2 25763 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 9-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  E  e.  Fin  /\  N  e.  V )  ->  (
 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  0 ) )
 
Theoremvdgn0frgrav2 25764 A vertex in a friendship graph with more than one vertex cannot have degree 0. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  ( 1  <  ( # `
  V )  ->  ( ( V VDeg  E ) `  N )  =/=  0 ) )
 
Theoremvdn1frgrav2 25765 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 10-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  E  e.  Fin  /\  N  e.  V )  ->  (
 1  <  ( # `  V )  ->  ( ( V VDeg 
 E ) `  N )  =/=  1 ) )
 
Theoremvdgn1frgrav2 25766 Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  ( 1  <  ( # `
  V )  ->  ( ( V VDeg  E ) `  N )  =/=  1 ) )
 
Theoremvdgfrgragt2 25767 Any vertex in a friendship graph (with more than one vertex - then, actually, the graph must have at least three vertices, because otherwise, it would not be a friendship graph) has at least degree 2, see remark 3 in [MertziosUnger] p. 153 (after Proposition 1): "It follows that deg(v) >= 2 for every node v of a friendship graph". (Contributed by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  N  e.  V )  ->  ( 1  <  ( # `
  V )  -> 
 2  <_  ( ( V VDeg  E ) `  N ) ) )
 
Theoremvdgn1frgrav3 25768* Any vertex in a friendship graph does not have degree 1, see remark 2 in [MertziosUnger] p. 153 (after Proposition 1): "... no node v of it [a friendship graph] may have deg(v) = 1.". (Contributed by Alexander van der Vekens, 4-Sep-2018.)
 |-  ( ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =/=  1 )
 
Theoremusgn0fidegnn0 25769* In a nonempty, finite graph there is a vertex having a nonnegative integer as degree. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  E. v  e.  V  E. n  e. 
 NN0  ( ( V VDeg 
 E ) `  v
 )  =  n )
 
16.3.4  Huneke's Proof of the Friendship Theorem

In this section, the friendship theorem friendship 25862 is proven by formalizing Huneke's proof, see [Huneke] pp. 1-2. The three claims (see frgrancvvdgeq 25783, frgraregorufr 25793 and frgregordn0 25810) and additional statements (numbered in the order of their occurence in the paper) in Huneke's proof are cited in the corresponding theorems.

 
Theoremfrgrancvvdeqlem1 25770* Lemma 1 for frgrancvvdeq 25782. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  Y  e.  ( V  \  { x } ) )
 
Theoremfrgrancvvdeqlem2 25771* Lemma 2 for frgrancvvdeq 25782. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  X  e/  N )
 
Theoremfrgrancvvdeqlem3 25772* Lemma 3 for frgrancvvdeq 25782. In a friendship graph, for each neighbor of a vertex there is exacly one neighbor of another vertex so that there is an edge between these two neighbors. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  E! y  e.  N  { x ,  y }  e.  ran  E )
 
Theoremfrgrancvvdeqlem4 25773* Lemma 4 for frgrancvvdeq 25782. The restricted iota of a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 18-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { ( iota_
 y  e.  N  { x ,  y }  e.  ran  E ) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N ) )
 
Theoremfrgrancvvdeqlem5 25774* Lemma 5 for frgrancvvdeq 25782. The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D
 --> N )
 
Theoremfrgrancvvdeqlem6 25775* Lemma 6 for frgrancvvdeq 25782. The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { ( A `  x ) }  =  ( ( <. V ,  E >. Neighbors  x )  i^i  N ) )
 
Theoremfrgrancvvdeqlem7 25776* Lemma 7 for frgrancvvdeq 25782. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ( ph  /\  x  e.  D )  ->  { x ,  ( A `  x ) }  e.  ran  E )
 
TheoremfrgrancvvdeqlemA 25777* Lemma A for frgrancvvdeq 25782. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A. x  e.  D  ( A `  x )  =/=  X )
 
TheoremfrgrancvvdeqlemB 25778* Lemma B for frgrancvvdeq 25782. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D -1-1-> ran  A )
 
TheoremfrgrancvvdeqlemC 25779* Lemma C for frgrancvvdeq 25782. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D -onto-> N )
 
Theoremfrgrancvvdeqlem8 25780* Lemma 8 for frgrancvvdeq 25782. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  D  =  ( <. V ,  E >. Neighbors  X )   &    |-  N  =  ( <. V ,  E >. Neighbors  Y )   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  ( ph  ->  Y  e/  D )   &    |-  ( ph  ->  V FriendGrph  E )   &    |-  A  =  ( x  e.  D  |->  (
 iota_ y  e.  N  { x ,  y }  e.  ran  E ) )   =>    |-  ( ph  ->  A : D
 -1-1-onto-> N )
 
Theoremfrgrancvvdeqlem9 25781* Lemma 9 for frgrancvvdeq 25782. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x } )
 ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  E. f  f : ( <. V ,  E >. Neighbors  x ) -1-1-onto-> ( <. V ,  E >. Neighbors  y ) ) )
 
Theoremfrgrancvvdeq 25782* In a finite friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( ( V FriendGrph  E  /\  E  e.  Fin )  ->  A. x  e.  V  A. y  e.  ( V 
 \  { x }
 ) ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  ( ( V VDeg  E ) `  x )  =  (
 ( V VDeg  E ) `  y ) ) )
 
Theoremfrgrancvvdgeq 25783* In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y, are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  A. x  e.  V  A. y  e.  ( V  \  { x } )
 ( y  e/  ( <. V ,  E >. Neighbors  x )  ->  ( ( V VDeg 
 E ) `  x )  =  ( ( V VDeg  E ) `  y
 ) ) )
 
Theoremfrgrawopreglem1 25784* Lemma 1 for frgrawopreg 25789. In a friendship graph, the classes A and B are sets. The definition of A and B corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( V FriendGrph  E  ->  ( A  e.  _V  /\  B  e.  _V )
 )
 
Theoremfrgrawopreglem2 25785* Lemma 2 for frgrawopreg 25789. In a friendship graph with at least two vertices, the degree of a vertex must be at least 2. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( ( V FriendGrph  E  /\  1  <  ( # `
  V )  /\  A  =/=  (/) )  ->  1  <  K )
 
Theoremfrgrawopreglem3 25786* Lemma 3 for frgrawopreg 25789. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( ( X  e.  A  /\  Y  e.  B )  ->  (
 ( V VDeg  E ) `  X )  =/=  (
 ( V VDeg  E ) `  Y ) )
 
Theoremfrgrawopreglem4 25787* Lemma 4 for frgrawopreg 25789. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( V FriendGrph  E  ->  A. a  e.  A  A. b  e.  B  { a ,  b }  e.  ran  E )
 
Theoremfrgrawopreglem5 25788* Lemma 5 for frgrawopreg 25789. If A as well as B contain at least two vertices in a friendship graph, there is a 4-cycle in the graph. This corresponds to statement 6 in [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( ( V FriendGrph  E  /\  1  <  ( # `
  A )  /\  1  <  ( # `  B ) )  ->  E. a  e.  A  E. x  e.  A  E. b  e.  B  E. y  e.  B  ( ( b  =/=  y  /\  a  =/=  x )  /\  ( { a ,  b }  e.  ran  E  /\  { x ,  b }  e.  ran  E )  /\  ( { a ,  y }  e.  ran  E  /\  { x ,  y }  e.  ran  E ) ) )
 
Theoremfrgrawopreg 25789* In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( V FriendGrph  E  ->  ( ( ( # `  A )  =  1  \/  A  =  (/) )  \/  ( ( # `  B )  =  1  \/  B  =  (/) ) ) )
 
Theoremfrgrawopreg1 25790* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( ( V FriendGrph  E  /\  ( # `  A )  =  1 )  ->  E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E )
 
Theoremfrgrawopreg2 25791* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  A  =  { x  e.  V  |  ( ( V VDeg  E ) `  x )  =  K }   &    |-  B  =  ( V 
 \  A )   =>    |-  ( ( V FriendGrph  E  /\  ( # `  B )  =  1 )  ->  E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E )
 
Theoremfrgraregorufr0 25792* In a friendship graph there are either no vertices having degree  K, or all vertices have degree 
K for any (nonnegative integer)  K, unless there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( V FriendGrph  E  ->  (
 A. v  e.  V  ( ( V VDeg  E ) `  v )  =  K  \/  A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =/=  K  \/  E. v  e.  V  A. w  e.  ( V  \  { v } ) { v ,  w }  e.  ran  E ) )
 
Theoremfrgraregorufr 25793* If there is a vertex having degree 
K for each (nonnegative integer)  K in a friendship graph, then either all vertices have degree  K or there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
 |-  ( V FriendGrph  E  ->  ( E. a  e.  V  ( ( V VDeg  E ) `  a )  =  K  ->  ( A. v  e.  V  (
 ( V VDeg  E ) `  v )  =  K  \/  E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E ) ) )
 
Theoremfrgraeu 25794* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b ( { A ,  b }  e.  ran  E 
 /\  { b ,  C }  e.  ran  E ) ) )
 
Theoremfrg2woteu 25795* For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  A  =/=  B )  ->  E! c  e.  V  <. A ,  c ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) )
 
Theoremfrg2wotn0 25796 In a friendship graph, there is always a path/walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  A  =/=  B )  ->  ( A ( V 2WalksOnOt  E ) B )  =/=  (/) )
 
Theoremfrg2wot1 25797 In a friendship graph, there is exactly one walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  A  =/=  B )  ->  ( # `  ( A ( V 2WalksOnOt  E ) B ) )  =  1 )
 
Theoremfrg2spot1 25798 In a friendship graph, there is exactly one simple path of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
 |-  ( ( V FriendGrph  E  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  A  =/=  B )  ->  ( # `  ( A ( V 2SPathOnOt  E ) B ) )  =  1 )
 
Theoremfrg2woteqm 25799 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 20-Feb-2018.)
 |-  ( ( V FriendGrph  E  /\  A  =/=  B )  ->  ( ( <. A ,  P ,  B >.  e.  ( A ( V 2WalksOnOt  E ) B ) 
 /\  <. B ,  Q ,  A >.  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  Q  =  P )
 )
 
Theoremfrg2woteq 25800 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)
 |-  ( ( V FriendGrph  E  /\  A  =/=  B )  ->  ( ( P  e.  ( A ( V 2WalksOnOt  E ) B )  /\  Q  e.  ( B ( V 2WalksOnOt  E ) A ) )  ->  ( ( 1st `  ( 1st `  P ) )  =  ( 2nd `  Q )  /\  ( 2nd `  ( 1st `  P ) )  =  ( 2nd `  ( 1st `  P ) ) 
 /\  ( 1st `  ( 1st `  Q ) )  =  ( 2nd `  P ) ) ) )
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