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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeupath 25701* 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 25702 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 25703* 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 25704* 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 25705* 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 25706* 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 25707 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 25701 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 25709 as special undirected simple graphs without loops (see frisusgra 25712). In subsection "The friendship theorem for small graphs", the friendship theorem for small graphs (with up to 3 vertices) is proved, see 1to3vfriendship 25728. The general friendship theorem friendship 25842 ( |-  ( ( 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 25709, a graph without vertices is a friendship graph (see frgra0 25714), 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 3777).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 25718, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 25719, 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 25722, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 25727 (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 25730 and n4cyclfrgra 25738 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.).

 
16.3.1  Friendship graphs - basics
 
Syntaxcfrgra 25708 Extend class notation with Friendship Graphs.
 class FriendGrph
 
Definitiondf-frgra 25709* 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 25710* 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 25711* 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 25712 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 25713 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 25714 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (/) FriendGrph  (/)
 
Theoremfrgraunss 25715* 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 25716* 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 25717* 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 25718 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 25719 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 25720* Lemma 1 for frgra3v 25722. (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 25721* Lemma 2 for frgra3v 25722. (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 25722 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 25723* 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 25724* Lemma for 3vfriswmgra 25725. (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 25725* 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 25726* 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 25727* 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 25728* 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 25729* 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 25730* 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 25731* 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 25732* 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 25733* 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 25734* 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 25735* 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 25736 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 25737* 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 25738 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 25739 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 25740 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 25741 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 25742 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 25743 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 25744 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 25745 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 25746 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 25747 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 25748* 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 25749* 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 25842 is proven by formalizing Huneke's proof, see [Huneke] pp. 1-2. The three claims (see frgrancvvdgeq 25763, frgraregorufr 25773 and frgregordn0 25790) and additional statements (numbered in the order of their occurence in the paper) in Huneke's proof are cited in the corresponding theorems.

 
Theoremfrgrancvvdeqlem1 25750* Lemma 1 for frgrancvvdeq 25762. (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 25751* Lemma 2 for frgrancvvdeq 25762. (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 25752* Lemma 3 for frgrancvvdeq 25762. 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 25753* Lemma 4 for frgrancvvdeq 25762. 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 25754* Lemma 5 for frgrancvvdeq 25762. 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 25755* Lemma 6 for frgrancvvdeq 25762. 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 25756* Lemma 7 for frgrancvvdeq 25762. (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 25757* Lemma A for frgrancvvdeq 25762. 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 25758* Lemma B for frgrancvvdeq 25762. 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 25759* Lemma C for frgrancvvdeq 25762. 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 25760* Lemma 8 for frgrancvvdeq 25762. (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 25761* Lemma 9 for frgrancvvdeq 25762. (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 25762* 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 25763* 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 25764* Lemma 1 for frgrawopreg 25769. 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 25765* Lemma 2 for frgrawopreg 25769. 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 25766* Lemma 3 for frgrawopreg 25769. 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 25767* Lemma 4 for frgrawopreg 25769. 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 25768* Lemma 5 for frgrawopreg 25769. 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 25769* 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 25770* 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 25771* 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 25772* 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 25773* 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 25774* 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 25775* 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 25776 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 25777 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 25778 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 25779 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 25780 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 ) ) ) )
 
Theorem2spotdisj 25781* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  A  e.  V )  -> Disj  b  e.  ( V  \  { A } ) ( A ( V 2SPathOnOt  E )
 b ) )
 
Theorem2spotiundisj 25782* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  -> Disj  a  e.  V  U_ b  e.  ( V  \  { a } )
 ( a ( V 2SPathOnOt  E ) b ) )
 
Theoremfrghash2spot 25783 The number of simple paths of length 2 is n*(n-1) in a friendship graph with  n vertices. This corresponds to the proof of claim 3 in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, the order of vertices is not respected by Huneke, so he only counts half of the paths which are existing when respecting the order as it is the case for simple paths represented by ordered triples. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
 |-  ( ( V FriendGrph  E  /\  ( V  e.  Fin  /\  V  =/=  (/) ) ) 
 ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V )  x.  ( ( # `  V )  -  1
 ) ) )
 
Theorem2spot0 25784 If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( ( V  =  (/)  /\  E  e.  X ) 
 ->  ( V 2SPathOnOt  E )  =  (/) )
 
Theoremusg2spot2nb 25785* The set of paths of length 2 with a given vertex in the middle for a finite graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) }
 )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( M `  N )  =  U_ x  e.  ( <. V ,  E >. Neighbors  N ) U_ y  e.  ( ( <. V ,  E >. Neighbors  N )  \  { x } ) { <. x ,  N ,  y >. } )
 
Theoremusgreghash2spotv 25786* According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For simple paths of length 2 represented by ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) }
 )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  A. v  e.  V  ( ( ( V VDeg 
 E ) `  v
 )  =  K  ->  ( # `  ( M `  v ) )  =  ( K  x.  ( K  -  1 ) ) ) )
 
Theoremusgreg2spot 25787* In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) }
 )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( A. v  e.  V  ( ( V VDeg 
 E ) `  v
 )  =  K  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  ( M `  x ) ) )
 
Theorem2spotmdisj 25788* The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  M  =  ( a  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  |  ( t  e.  ( V 2SPathOnOt  E )  /\  ( 2nd `  ( 1st `  t ) )  =  a ) }
 )   =>    |-  ( V  e.  _V  -> Disj  x  e.  V  ( M `  x ) )
 
Theoremusgreghash2spot 25789* In a finite k-regular graph with N vertices there are N times " k choose 2 " paths with length 2, according to statement 8 in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by ordered triples, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v )  =  K  ->  ( # `  ( V 2SPathOnOt  E ) )  =  ( ( # `  V )  x.  ( K  x.  ( K  -  1
 ) ) ) ) )
 
Theoremfrgregordn0 25790* If a nonempty friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/= 
 (/) )  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v )  =  K  ->  ( # `  V )  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
 
Theoremfrrusgraord 25791 If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frgregordn0 25790, using the definition RegUSGrph (df-rusgra 25645). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
 |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( ( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( # `  V )  =  ( ( K  x.  ( K  -  1 ) )  +  1 ) ) )
 
Theoremfrgraregorufrg 25792* If there is a vertex having degree  k for each nonnegative integer  k in a friendship graph, then 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". Variant of frgraregorufr 25773 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
 |-  ( V FriendGrph  E  ->  A. k  e.  NN0  ( E. a  e.  V  ( ( V VDeg  E ) `  a )  =  k  ->  ( <. V ,  E >. RegUSGrph  k  \/ 
 E. v  e.  V  A. w  e.  ( V 
 \  { v }
 ) { v ,  w }  e.  ran  E ) ) )
 
Theoremnumclwlk3lem3 25793 Lemma 3 for numclwwlk3 25829. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
 |-  ( ( K  e.  CC  /\  Y  e.  CC  /\  N  e.  ( ZZ>= `  2 ) )  ->  ( ( ( K ^ ( N  -  2 ) )  -  Y )  +  ( K  x.  Y ) )  =  ( ( ( K  -  1 )  x.  Y )  +  ( K ^ ( N  -  2 ) ) ) )
 
Theoremextwwlkfablem1 25794 Lemma 1 for extwwlkfab 25810. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
 |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
 `  2 ) ) 
 /\  w  e.  (
 ( V ClWWalksN  E ) `  N ) )  /\  ( ( w `  0 )  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `  0 ) ) ) 
 ->  ( w `  ( N  -  1 ) )  e.  ( <. V ,  E >. Neighbors  X ) )
 
Theoremextwwlkfablem2lem 25795 Lemma for extwwlkfablem2 25798. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
 |-  ( ( w  e. Word  V  /\  ( # `  w )  =  N  /\  N  e.  ( ZZ>= `  2 ) )  ->  ( # `  ( w substr  <. 0 ,  ( N  -  2 ) >. ) )  =  ( N  -  2 ) )
 
Theoremclwwlkextfrlem1 25796 Lemma for numclwwlk2lem1 25822. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  ( ( ( X  e.  V  /\  N  e.  NN  /\  Z  e.  V )  /\  ( W  e.  ( ( V WWalksN  E ) `  N )  /\  ( W `  0 )  =  X  /\  ( lastS  `  W )  =/=  X ) ) 
 ->  ( ( ( W ++ 
 <" Z "> ) `  0 )  =  X  /\  ( ( W ++  <" Z "> ) `  N )  =/=  X ) )
 
Theoremnumclwwlkfvc 25797* Value of function  C, mapping a nonnegative number n to the closed walks having length n. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   =>    |-  ( N  e.  NN0  ->  ( C `  N )  =  ( ( V ClWWalksN  E ) `  N ) )
 
Theoremextwwlkfablem2 25798* Lemma 2 for extwwlkfab 25810. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   =>    |-  ( ( ( ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>=
 `  3 ) ) 
 /\  w  e.  (
 ( V ClWWalksN  E ) `  N ) )  /\  ( ( w `  0 )  =  X  /\  ( w `  ( N  -  2 ) )  =  ( w `  0 ) ) ) 
 ->  ( w substr  <. 0 ,  ( N  -  2
 ) >. )  e.  ( C `  ( N  -  2 ) ) )
 
Theoremnumclwwlkun 25799* The set of closed walks in an undirected simple graph is the union of the numbers of closed walks starting at each of the vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   =>    |-  ( ( V USGrph  E  /\  N  e.  NN0 )  ->  ( C `  N )  =  U_ x  e.  V  { w  e.  ( C `  N )  |  ( w `  0 )  =  x } )
 
Theoremnumclwwlkdisj 25800* The sets of closed walks starting at different vertices in an undirected simple graph are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   =>    |- Disj  x  e.  V  { w  e.  ( C `  N )  |  ( w `  0 )  =  x }
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