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Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremvtxdgfisf 39701 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.) (Revised by AV, 11-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  A  =  dom  I   =>    |-  ( ( G  e.  W  /\  A  e.  Fin )  ->  (VtxDeg `  G ) : V --> NN0 )
 
Theoremvtxdeqd 39702 Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)
 |-  ( ph  ->  G  e.  X )   &    |-  ( ph  ->  H  e.  Y )   &    |-  ( ph  ->  (Vtx `  H )  =  (Vtx `  G ) )   &    |-  ( ph  ->  (iEdg `  H )  =  (iEdg `  G ) )   =>    |-  ( ph  ->  (VtxDeg `  H )  =  (VtxDeg `  G ) )
 
Theoremvtxduhgr0e 39703 The degree of a vertex in an empty hypergraph is zero, because there are no edges. Analogue of vtxdg0e 39699. (Contributed by AV, 15-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  U  e.  V  /\  E  =  (/) )  ->  ( (VtxDeg `  G ) `  U )  =  0 )
 
Theoremvtxdun 39704 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.) (Revised by AV, 19-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  J  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  ( dom  I  i^i  dom  J )  =  (/) )   &    |-  ( ph  ->  Fun 
 I )   &    |-  ( ph  ->  Fun 
 J )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )   =>    |-  ( ph  ->  ( (VtxDeg `  U ) `  N )  =  ( ( (VtxDeg `  G ) `  N ) +e ( (VtxDeg `  H ) `  N ) ) )
 
Theoremvtxdfiun 39705 The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 19-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  J  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  ( dom  I  i^i  dom  J )  =  (/) )   &    |-  ( ph  ->  Fun 
 I )   &    |-  ( ph  ->  Fun 
 J )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )   &    |-  ( ph  ->  dom  I  e.  Fin )   &    |-  ( ph  ->  dom 
 J  e.  Fin )   =>    |-  ( ph  ->  ( (VtxDeg `  U ) `  N )  =  ( ( (VtxDeg `  G ) `  N )  +  ( (VtxDeg `  H ) `  N ) ) )
 
Theoremvtxduhgrun 39706 The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  J  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  ( dom  I  i^i  dom  J )  =  (/) )   &    |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  H  e. UHGraph  )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )   =>    |-  ( ph  ->  ( (VtxDeg `  U ) `  N )  =  ( ( (VtxDeg `  G ) `  N ) +e ( (VtxDeg `  H ) `  N ) ) )
 
Theoremvtxduhgrfiun 39707 The degree of a vertex in the union of two hypergraphs of finite size on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.) (Revised by AV, 7-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
 |-  I  =  (iEdg `  G )   &    |-  J  =  (iEdg `  H )   &    |-  V  =  (Vtx `  G )   &    |-  ( ph  ->  (Vtx `  H )  =  V )   &    |-  ( ph  ->  (Vtx `  U )  =  V )   &    |-  ( ph  ->  ( dom  I  i^i  dom  J )  =  (/) )   &    |-  ( ph  ->  G  e. UHGraph  )   &    |-  ( ph  ->  H  e. UHGraph  )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )   &    |-  ( ph  ->  dom  I  e.  Fin )   &    |-  ( ph  ->  dom 
 J  e.  Fin )   =>    |-  ( ph  ->  ( (VtxDeg `  U ) `  N )  =  ( ( (VtxDeg `  G ) `  N )  +  ( (VtxDeg `  H ) `  N ) ) )
 
Theoremvtxdlfgrval 39708* The value of the vertex degree function for a loop-free graph  G. (Contributed by AV, 23-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  A  =  dom  I   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( ( I : A
 --> { x  e.  ~P V  |  2  <_  ( # `  x ) }  /\  U  e.  V ) 
 ->  ( D `  U )  =  ( # `  { x  e.  A  |  U  e.  ( I `  x ) } ) )
 
Theoremvtxdumgrval 39709* The value of the vertex degree function for a multigraph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 23-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  A  =  dom  I   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( ( G  e. UMGraph  /\  U  e.  V ) 
 ->  ( D `  U )  =  ( # `  { x  e.  A  |  U  e.  ( I `  x ) } ) )
 
Theoremvtxdusgrval 39710* The value of the vertex degree function for a simple graph. (Contributed by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 11-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  A  =  dom  I   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( ( G  e. USGraph  /\  U  e.  V ) 
 ->  ( D `  U )  =  ( # `  { x  e.  A  |  U  e.  ( I `  x ) } ) )
 
Theoremvtxd0nedgb 39711* A vertex has degree 0 iff there is no edge incident with the vertex. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 22-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( U  e.  V  ->  ( ( D `  U )  =  0  <->  -.  E. i  e. 
 dom  I  U  e.  ( I `  i ) ) )
 
Theoremvtxdushgrfvedglem 39712* Lemma for vtxdushgrfvedg 39713 and vtxdusgrfvedg 39714. (Contributed by AV, 12-Dec-2020.) TODO-AV: proof can be shortened by using "bj-eleq2w", after it is moved to main.set.
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USHGraph  /\  U  e.  V )  ->  ( # `
  { i  e. 
 dom  (iEdg `  G )  |  U  e.  (
 (iEdg `  G ) `  i ) } )  =  ( # `  { e  e.  E  |  U  e.  e } ) )
 
Theoremvtxdushgrfvedg 39713* The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. USHGraph  /\  U  e.  V )  ->  ( D `  U )  =  ( ( # `  { e  e.  E  |  U  e.  e } ) +e
 ( # `  { e  e.  E  |  e  =  { U } }
 ) ) )
 
Theoremvtxdusgrfvedg 39714* The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  ( D `  U )  =  ( # `  { e  e.  E  |  U  e.  e } ) )
 
Theoremvtxduhgr0nedg 39715* If a vertex in a hypergraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 15-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  U  e.  V  /\  ( D `
  U )  =  0 )  ->  -.  E. v  e.  V  { U ,  v }  e.  E )
 
Theoremvtxdumgr0nedg 39716* If a vertex in a multigraph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 15-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  U  e.  V  /\  ( D `
  U )  =  0 )  ->  -.  E. v  e.  V  { U ,  v }  e.  E )
 
Theoremvtxduhgr0edgnel 39717* A vertex in a hypergraph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 24-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  U  e.  V )  ->  (
 ( D `  U )  =  0  <->  -.  E. e  e.  E  U  e.  e
 ) )
 
Theoremvtxdusgr0edgnel 39718* A vertex in a simple graph has degree 0 iff there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  (
 ( D `  U )  =  0  <->  -.  E. e  e.  E  U  e.  e
 ) )
 
Theoremvtxdusgr0edgnelALT 39719* Alternate proof of vtxdusgr0edgnel 39718, not based on vtxduhgr0edgnel 39717. A vertex in a simple graph has degree 0 if there is no edge incident with this vertex. (Contributed by AV, 17-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  (
 ( D `  U )  =  0  <->  -.  E. e  e.  E  U  e.  e
 ) )
 
Theoremvtxdgfusgrf 39720 The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. FinUSGraph  ->  (VtxDeg `  G ) : V --> NN0 )
 
Theoremvtxdgfusgr 39721* In a finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 12-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. FinUSGraph  ->  A. v  e.  V  ( (VtxDeg `  G ) `  v )  e. 
 NN0 )
 
Theorem1loopgruspgr 39722 A graph with one edge which is a loop is a simple pseudograph (see also uspgr1v1eop 39488). (Contributed by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )   =>    |-  ( ph  ->  G  e. USPGraph  )
 
Theorem1loopgredg 39723 The set of edges in a graph (simple pseudograph) with one edge which is a loop is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )   =>    |-  ( ph  ->  (Edg `  G )  =  { { N } } )
 
Theorem1loopgrnb0 39724 In a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )   =>    |-  ( ph  ->  ( G NeighbVtx  N )  =  (/) )
 
Theorem1loopgrvd2 39725 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop, the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )   =>    |-  ( ph  ->  ( (VtxDeg `  G ) `  N )  =  2 )
 
Theorem1loopgrvd0 39726 The vertex degree of a one-edge graph, case 1 (for a loop): a loop at a vertex other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  N  e.  V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )   &    |-  ( ph  ->  K  e.  ( V  \  { N }
 ) )   =>    |-  ( ph  ->  (
 (VtxDeg `  G ) `  K )  =  0
 )
 
Theorem1hevtxdg0 39727 The vertex degree of vertex  D in a graph  G with only one hyperedge  E is 0 if  D is not incident with the edge  E. (Contributed by AV, 2-Mar-2021.)
 |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  E >. } )   &    |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  E  e.  Y )   &    |-  ( ph  ->  D  e/  E )   =>    |-  ( ph  ->  (
 (VtxDeg `  G ) `  D )  =  0
 )
 
Theorem1hevtxdg1 39728 The vertex degree of vertex  D in a graph  G with only one hyperedge  E (not being a loop) is 1 if  D is incident with the edge  E. (Contributed by AV, 2-Mar-2021.)
 |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  E >. } )   &    |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  E  e.  ~P V )   &    |-  ( ph  ->  D  e.  E )   &    |-  ( ph  ->  2 
 <_  ( # `  E ) )   =>    |-  ( ph  ->  (
 (VtxDeg `  G ) `  D )  =  1
 )
 
Theorem1hegrlfgr 39729* --- TODO-AV: not used anymore!? ! ------------------------- A graph  G with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  E  e.  ~P V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  E >. } )   &    |-  ( ph  ->  { B ,  C }  C_  E )   =>    |-  ( ph  ->  (iEdg `  G ) : { A } --> { x  e.  ~P V  |  2  <_  ( # `  x ) }
 )
 
Theorem1hegrvtxdg1 39730 The vertex degree of a graph with one hyperedge, 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.) (Revised by AV, 23-Feb-2021.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  E  e.  ~P V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  E >. } )   &    |-  ( ph  ->  { B ,  C }  C_  E )   &    |-  ( ph  ->  (Vtx `  G )  =  V )   =>    |-  ( ph  ->  ( (VtxDeg `  G ) `  B )  =  1 )
 
Theorem1hegrvtxdg1r 39731 The vertex degree of a graph with one hyperedge, 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.) (Revised by AV, 23-Feb-2021.)
 |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  E  e.  ~P V )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  E >. } )   &    |-  ( ph  ->  { B ,  C }  C_  E )   &    |-  ( ph  ->  (Vtx `  G )  =  V )   =>    |-  ( ph  ->  ( (VtxDeg `  G ) `  C )  =  1 )
 
Theorem1egrvtxdg1 39732 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.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  C } >. } )   =>    |-  ( ph  ->  ( (VtxDeg `  G ) `  B )  =  1 )
 
Theorem1egrvtxdg1r 39733 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.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  C } >. } )   =>    |-  ( ph  ->  ( (VtxDeg `  G ) `  C )  =  1 )
 
Theorem1egrvtxdg0 39734 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.) (Revised by AV, 21-Feb-2021.)
 |-  ( ph  ->  (Vtx `  G )  =  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  B  =/=  C )   &    |-  ( ph  ->  D  e.  V )   &    |-  ( ph  ->  C  =/=  D )   &    |-  ( ph  ->  (iEdg `  G )  =  { <. A ,  { B ,  D } >. } )   =>    |-  ( ph  ->  (
 (VtxDeg `  G ) `  C )  =  0
 )
 
Theoremp1evtxdeqlem 39735 Lemma for p1evtxdeq 39736 and p1evtxdp1 39737. (Contributed by AV, 3-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  (Vtx `  F )  =  V )   &    |-  ( ph  ->  (iEdg `  F )  =  ( I  u.  { <. K ,  E >. } ) )   &    |-  ( ph  ->  K  e.  X )   &    |-  ( ph  ->  K  e/  dom  I )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  E  e.  Y )   =>    |-  ( ph  ->  (
 (VtxDeg `  F ) `  U )  =  (
 ( (VtxDeg `  G ) `  U ) +e ( (VtxDeg `  <. V ,  { <. K ,  E >. } >. ) `  U ) ) )
 
Theoremp1evtxdeq 39736 If an edge  E which does not contain vertex  U is added to a graph  G (yielding a graph  F), the degree of  U is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  (Vtx `  F )  =  V )   &    |-  ( ph  ->  (iEdg `  F )  =  ( I  u.  { <. K ,  E >. } ) )   &    |-  ( ph  ->  K  e.  X )   &    |-  ( ph  ->  K  e/  dom  I )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  E  e.  Y )   &    |-  ( ph  ->  U 
 e/  E )   =>    |-  ( ph  ->  ( (VtxDeg `  F ) `  U )  =  ( (VtxDeg `  G ) `  U ) )
 
Theoremp1evtxdp1 39737 If an edge  E (not being a loop) which contains vertex  U is added to a graph  G (yielding a graph  F), the degree of  U is increased by 1. (Contributed by AV, 3-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  Fun  I )   &    |-  ( ph  ->  (Vtx `  F )  =  V )   &    |-  ( ph  ->  (iEdg `  F )  =  ( I  u.  { <. K ,  E >. } ) )   &    |-  ( ph  ->  K  e.  X )   &    |-  ( ph  ->  K  e/  dom  I )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  E  e.  ~P V )   &    |-  ( ph  ->  U  e.  E )   &    |-  ( ph  ->  2  <_  ( # `  E ) )   =>    |-  ( ph  ->  (
 (VtxDeg `  F ) `  U )  =  (
 ( (VtxDeg `  G ) `  U ) +e 1 ) )
 
Theoremuspgrloopvtx 39738 The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488). (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( V  e.  W  ->  (Vtx `  G )  =  V )
 
Theoremuspgrloopvtxel 39739 A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488). (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( ( V  e.  W  /\  N  e.  V )  ->  N  e.  (Vtx `  G ) )
 
Theoremuspgrloopiedg 39740 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488) is a singleton of a singleton. (Contributed by AV, 21-Feb-2021.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  X )  ->  (iEdg `  G )  =  { <. A ,  { N } >. } )
 
Theoremuspgrloopedg 39741 The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  X )  ->  (Edg `  G )  =  { { N } } )
 
Theoremuspgrloopnb0 39742 In a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488), the vertex connected with itself by the loop has no neighbors. (Contributed by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  X  /\  N  e.  V ) 
 ->  ( G NeighbVtx  N )  =  (/) )
 
Theoremuspgrloopvd2 39743 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. I. e. in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 39488), the vertex connected with itself by the loop has degree 2. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 21-Feb-2021.)
 |-  G  =  <. V ,  { <. A ,  { N } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  X  /\  N  e.  V ) 
 ->  ( (VtxDeg `  G ) `  N )  =  2 )
 
Theoremumgr2v2evtx 39744 The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( V  e.  W  ->  (Vtx `  G )  =  V )
 
Theoremumgr2v2evtxel 39745 A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  V )  ->  A  e.  (Vtx `  G ) )
 
Theoremumgr2v2eiedg 39746 The edge function in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V ) 
 ->  (iEdg `  G )  =  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } )
 
Theoremumgr2v2eedg 39747 The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V ) 
 ->  (Edg `  G )  =  { { A ,  B } } )
 
Theoremumgr2v2e 39748 A multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  ->  G  e. UMGraph  )
 
Theoremumgr2v2enb1 39749 In a multigraph with two edges connecting the same two vertices, each of the vertices has one neighbor. (Contributed by AV, 18-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  ->  ( G NeighbVtx  A )  =  { B } )
 
Theoremumgr2v2evd2 39750 In a multigraph with two edges connecting the same two vertices, each of the vertices has degree 2. (Contributed by AV, 18-Dec-2020.)
 |-  G  =  <. V ,  { <. 0 ,  { A ,  B } >. ,  <. 1 ,  { A ,  B } >. } >.   =>    |-  ( ( ( V  e.  W  /\  A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  ->  ( (VtxDeg `  G ) `  A )  =  2 )
 
Theoremhashnbusgrvd 39751 In a simple graph, the number of neighbors of a vertex is the degree of this vertex. This theorem does not hold for (simple) pseudographs, because a vertex connected with itself only by a loop has no neighbors, see uspgrloopnb0 39742, but degree 2, see uspgrloopvd2 39743. And it does not hold for multigraphs, because a vertex connected with only one other vertex by two edges has one neighbor, see umgr2v2enb1 39749, but also degree 2, see umgr2v2evd2 39750. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 15-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  ( # `
  ( G NeighbVtx  U ) )  =  ( (VtxDeg `  G ) `  U ) )
 
Theoremusgruvtxvdb 39752 In a finite simple graph with n vertices a vertex is universal iff the vertex has degree  n  -  1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  U  e.  V )  ->  ( U  e.  (UnivVtx `  G ) 
 <->  ( (VtxDeg `  G ) `  U )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremvdiscusgrb 39753* A finite simple graph with n vertices is complete iff every vertex has degree  n  -  1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. FinUSGraph  ->  ( G  e. ComplUSGraph  <->  A. v  e.  V  ( (VtxDeg `  G ) `  v )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremvdiscusgr 39754* In a finite complete simple graph with n vertices every vertex has degree  n  -  1. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G  e. FinUSGraph  ->  ( A. v  e.  V  (
 (VtxDeg `  G ) `  v )  =  (
 ( # `  V )  -  1 )  ->  G  e. ComplUSGraph ) )
 
Theoremvtxdusgradjvtx 39755* The degree of a vertex in a simple graphs is the number of vertices adjacent to this vertex. (Contributed by Alexander van der Vekens, 9-Jul-2018.) (Revised by AV, 23-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  (
 (VtxDeg `  G ) `  U )  =  ( # `
  { v  e.  V  |  { U ,  v }  e.  E } ) )
 
Theoremusgrvd0nedg 39756* 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, 20-Dec-2017.) (Revised by AV, 16-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. USGraph  /\  U  e.  V )  ->  (
 ( (VtxDeg `  G ) `  U )  =  0  ->  -.  E. v  e.  V  { U ,  v }  e.  E ) )
 
Theoremuhgrvd00 39757* If every vertex in a hypergraph has degree 0, there is no edge in the graph. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. UHGraph  ->  ( A. v  e.  V  (
 (VtxDeg `  G ) `  v )  =  0  ->  E  =  (/) ) )
 
Theoremusgrvd00 39758* 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.) (Revised by AV, 17-Dec-2020.) (Proof shortened by AV, 23-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. USGraph  ->  ( A. v  e.  V  (
 (VtxDeg `  G ) `  v )  =  0  ->  E  =  (/) ) )
 
Theoremvdegp1ai-av 39759* 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.) (Revised by AV, 3-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  U  e.  V   &    |-  I  =  (iEdg `  G )   &    |-  I  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }   &    |-  (
 (VtxDeg `  G ) `  U )  =  P   &    |-  (Vtx `  F )  =  V   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  Y  e.  V   &    |-  Y  =/=  U   &    |-  (iEdg `  F )  =  ( I ++  <" { X ,  Y } "> )   =>    |-  ( (VtxDeg `  F ) `  U )  =  P
 
Theoremvdegp1bi-av 39760* 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.) (Revised by AV, 3-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  U  e.  V   &    |-  I  =  (iEdg `  G )   &    |-  I  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }   &    |-  (
 (VtxDeg `  G ) `  U )  =  P   &    |-  (Vtx `  F )  =  V   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  (iEdg `  F )  =  ( I ++  <" { U ,  X } "> )   =>    |-  ( (VtxDeg `  F ) `  U )  =  ( P  +  1 )
 
Theoremvdegp1ci-av 39761* 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.) (Revised by AV, 3-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  U  e.  V   &    |-  I  =  (iEdg `  G )   &    |-  I  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 }   &    |-  (
 (VtxDeg `  G ) `  U )  =  P   &    |-  (Vtx `  F )  =  V   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  (iEdg `  F )  =  ( I ++  <" { X ,  U } "> )   =>    |-  ( (VtxDeg `  F ) `  U )  =  ( P  +  1 )
 
21.33.8.13  Regular graphs

With df-rgr 39764 and df-rusgr 39765, k-regularity of a (simple) graph is defined as predicate RegGraph resp. RegUSGraph. Instead of defining a predicate, an alternative could have been to define a function that maps an extended nonnegative integer to the class of "graphs" in which every vertex has the extended nonnegative integer as degree: RegGraph  =  ( k  e. NN0*  |->  { g  |  A. v  e.  (Vtx `  g )
( (VtxDeg `  g
) `  v )  =  k } ). This function, however, would not be defined for  k  =  0 (see rgrx0nd 39798), because  { g  |  A. v  e.  (Vtx
`  g ) ( (VtxDeg `  g ) `  v )  =  0 } is not a set (see rgrprcx 39796). It is expected that this function is not defined for every  k  e. NN0* (how could this be proven?).

 
Syntaxcrgr 39762 Extend class notation to include the class of all regular graphs.
 class RegGraph
 
Syntaxcrusgr 39763 Extend class notation to include the class of all regular simple graphs.
 class RegUSGraph
 
Definitiondf-rgr 39764* Define the "k-regular" predicate, which is true for a "graph" being k-regular: read  G RegGraph  K as " G is  K-regular" or " G is a  K-regular graph". Note that  K is allowed to be positive infinity ( K  e. NN0*), as proposed by GL. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |- RegGraph  =  { <. g ,  k >.  |  ( k  e. NN0*  /\  A. v  e.  (Vtx `  g
 ) ( (VtxDeg `  g
 ) `  v )  =  k ) }
 
Definitiondf-rusgr 39765* Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read  G RegUSGraph  K as  G is a  K-regular simple graph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 18-Dec-2020.)
 |- RegUSGraph  =  { <. g ,  k >.  |  ( g  e. USGraph  /\  g RegGraph  k ) }
 
Theoremisrgr 39766* The property of a class being a k-regular graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e.  W  /\  K  e.  Z ) 
 ->  ( G RegGraph  K  <->  ( K  e. NN0*  /\ 
 A. v  e.  V  ( D `  v )  =  K ) ) )
 
Theoremrgrprop 39767* The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( G RegGraph  K  ->  ( K  e. NN0*  /\  A. v  e.  V  ( D `  v )  =  K ) )
 
Theoremisrusgr 39768 The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.)
 |-  (
 ( G  e.  W  /\  K  e.  Z ) 
 ->  ( G RegUSGraph  K  <->  ( G  e. USGraph  /\  G RegGraph  K ) ) )
 
Theoremrusgrprop 39769 The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
 |-  ( G RegUSGraph  K  ->  ( G  e. USGraph 
 /\  G RegGraph  K ) )
 
Theoremrusgrrgr 39770 A k-regular simple graph is a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
 |-  ( G RegUSGraph  K  ->  G RegGraph  K )
 
Theoremrusgrusgr 39771 A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.)
 |-  ( G RegUSGraph  K  ->  G  e. USGraph  )
 
Theoremisrusgr0 39772* The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e.  W  /\  K  e.  Z ) 
 ->  ( G RegUSGraph  K  <->  ( G  e. USGraph  /\  K  e. NN0*  /\  A. v  e.  V  ( D `  v )  =  K ) ) )
 
Theoremrusgrprop0 39773* The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  ( G RegUSGraph  K  ->  ( G  e. USGraph 
 /\  K  e. NN0*  /\  A. v  e.  V  ( D `  v )  =  K ) )
 
Theoremusgreqdrusgr 39774* If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. USGraph  /\  K  e. NN0*  /\  A. v  e.  V  ( D `  v )  =  K )  ->  G RegUSGraph  K )
 
Theoremfusgrregdegfi 39775* In a nonempty finite simple graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 19-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  V  =/= 
 (/) )  ->  ( A. v  e.  V  ( D `  v )  =  K  ->  K  e.  NN0 ) )
 
Theoremfusgrn0eqdrusgr 39776* If all vertices in a nonempty finite simple graph have the same (finite) degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   &    |-  D  =  (VtxDeg `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  V  =/= 
 (/) )  ->  ( A. v  e.  V  ( D `  v )  =  K  ->  G RegUSGraph  K ) )
 
Theorem0edg0rgr 39777 A graph is 0-regular if it has no edges. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  (
 ( G  e.  W  /\  (iEdg `  G )  =  (/) )  ->  G RegGraph  0 )
 
Theoremuhgr0edg0rgr 39778 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
 |-  (
 ( G  e. UHGraph  /\  (Edg `  G )  =  (/) )  ->  G RegGraph  0 )
 
Theoremuhgr0edg0rgrb 39779 A hypergraph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 24-Dec-2020.)
 |-  ( G  e. UHGraph  ->  ( G RegGraph  0 
 <->  (Edg `  G )  =  (/) ) )
 
Theoremusgr0edg0rusgr 39780 A simple graph is 0-regular iff it has no edges. (Contributed by Alexander van der Vekens, 12-Jul-2018.) (Revised by AV, 19-Dec-2020.) (Proof shortened by AV, 24-Dec-2020.)
 |-  ( G  e. USGraph  ->  ( G RegUSGraph  0 
 <->  (Edg `  G )  =  (/) ) )
 
Theorem0vtxrgr 39781* A graph with no vertices is k-regular for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/) )  ->  A. k  e. NN0*  G RegGraph  k )
 
Theorem0vtxrusgr 39782* A graph with no vertices and an empty edge function is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  (
 ( G  e.  W  /\  (Vtx `  G )  =  (/)  /\  (iEdg `  G )  =  (/) )  ->  A. k  e. NN0*  G RegUSGraph  k )
 
Theorem0uhgrrusgr 39783* The null graph as hypergraph is a k-regular simple graph for every k. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  (
 ( G  e. UHGraph  /\  (Vtx `  G )  =  (/) )  ->  A. k  e. NN0*  G RegUSGraph  k )
 
Theorem0grrusgr 39784 The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.)
 |-  A. k  e. NN0* 
 (/) RegUSGraph  k
 
Theorem0grrgr 39785 The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
 |-  A. k  e. NN0* 
 (/) RegGraph  k
 
Theoremcusgrrusgr 39786 A complete simple graph with n vertices (at least one) is (n-1)-regular. (Contributed by Alexander van der Vekens, 10-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. ComplUSGraph  /\  V  e.  Fin  /\  V  =/=  (/) )  ->  G RegUSGraph  ( ( # `  V )  -  1 ) )
 
Theoremcusgrm1rusgr 39787 A finite simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGraph was allowed for  k  e.  ZZ, then the assumption  V  =/=  (/) could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. FinUSGraph  /\  V  =/= 
 (/) )  ->  ( G  e. ComplUSGraph  <->  G RegUSGraph  ( ( # `  V )  -  1
 ) ) )
 
Theoremrusgrpropnb 39788* The properties of a k-regular simple graph expressed with neighbors. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 26-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G RegUSGraph  K  ->  ( G  e. USGraph 
 /\  K  e. NN0*  /\  A. v  e.  V  ( # `
  ( G NeighbVtx  v ) )  =  K ) )
 
Theoremrusgrpropedg 39789* The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G RegUSGraph  K  ->  ( G  e. USGraph 
 /\  K  e. NN0*  /\  A. v  e.  V  ( # `
  { e  e.  (Edg `  G )  |  v  e.  e } )  =  K ) )
 
Theoremrusgrpropadjvtx 39790* The properties of a k-regular simple graph expressed with adjacent vertices. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 27-Dec-2020.)
 |-  V  =  (Vtx `  G )   =>    |-  ( G RegUSGraph  K  ->  ( G  e. USGraph 
 /\  K  e. NN0*  /\  A. v  e.  V  ( # `
  { k  e.  V  |  { v ,  k }  e.  (Edg `  G ) } )  =  K ) )
 
Theoremrusgr1vtxlem 39791* Lemma for rusgr1vtx 39792. (Contributed by AV, 27-Dec-2020.)
 |-  (
 ( ( A. v  e.  V  ( # `  A )  =  K  /\  A. v  e.  V  A  =  (/) )  /\  ( V  e.  W  /\  ( # `  V )  =  1 ) ) 
 ->  K  =  0 )
 
Theoremrusgr1vtx 39792 If a k-regular simple graph has only one vertex, then k must be  0. (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.)
 |-  (
 ( ( # `  (Vtx `  G ) )  =  1  /\  G RegUSGraph  K ) 
 ->  K  =  0 )
 
Theoremrgrusgrprc 39793* The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  { g  e. USGraph  |  A. v  e.  (Vtx `  g )
 ( (VtxDeg `  g
 ) `  v )  =  0 }  e/  _V
 
Theoremrusgrprc 39794 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  { g  |  g RegUSGraph  0 }  e/  _V
 
Theoremrgrprc 39795 The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  { g  |  g RegGraph  0 }  e/  _V
 
Theoremrgrprcx 39796* The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
 |-  { g  |  A. v  e.  (Vtx `  g ) ( (VtxDeg `  g ) `  v
 )  =  0 } 
 e/  _V
 
Theoremrgrx0ndm 39797* 0 is not in the domain of the potentially alternatively defined vertex degree function. (Contributed by AV, 28-Dec-2020.)
 |-  R  =  ( k  e. NN0*  |->  { g  |  A. v  e.  (Vtx `  g ) ( (VtxDeg `  g ) `  v
 )  =  k }
 )   =>    |-  0  e/  dom  R
 
Theoremrgrx0nd 39798* The potentially alternatively defined vertex degree function is not defined for 0. (Contributed by AV, 28-Dec-2020.)
 |-  R  =  ( k  e. NN0*  |->  { g  |  A. v  e.  (Vtx `  g ) ( (VtxDeg `  g ) `  v
 )  =  k }
 )   =>    |-  ( R `  0
 )  =  (/)
 
21.33.8.14  Walks

A "walk" within a graph is usually defined for simple graphs, multigraphs or even pseudographs as "alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. This definition requires the edges to connect two vertices at most (loops are also allowed: if e(i) is a loop, then x(i-1) = x(i)). For hypergraphs containing hyperedges (i.e. edges connecting more than two vertices), however, a more general definition is needed. Two approaches for a definition applicable for arbitrary hypergraphs are used in literature: "walks on the vertex level" and "walks on the edge level" (see Aksoy, Joslyn, Marrero, Praggastis, Purvine: "Hypernetwork science via high-order hypergraph walks", June 2020, https://doi.org/10.1140/epjds/s13688-020-00231-0):

"walks on the edge level": For a positive integer s, an s-walk of length k between hyperedges f and g is a sequence of hyperedges, f=e(0), e(1), ... , e(k)=g, where for j=1, ... , k, e(j-1) =/= e(j) and e(j-1) and e(j) have at least s vertices in common (according to Aksoy et al.).

"walks on the vertex level": For a positive integer s, an s-walk of length k between vertices a and b is a sequence of vertices, a=v(0), v(1), ... , v(k)=b, where for j=1, ... , k, v(j-1) and v(j) are connected by at least s edges (analogous to Aksoy et al.).

There are two imperfections for the definition for "walks on the edge level": one is that a walk of length 1 consists of two edges (or a walk of length 0 consists of one edge), whereas a walk of length 1 on the vertex level consists of two vertices and one edge (or a walk of length 0 consists of one vertex and no edge). The other is that edges, especially loops, can be traversed only once (and not repeatedly) because of the condition e(j-1) =/= e(j). The latter is avoided in the definition for EdgWalks, see df-ewlks 39803. To be compatible with the (usual) definition of walks for pseudographs, walks also suitable for arbitrary hypergraphs are defined "on the vertex level" in the following as 1Walks, see df-1wlks 39804, restricting s to 1. 1wlk1ewlk 39839 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level".

 
Syntaxcewlks 39799 Extend class notation with s-walks "on the edge level" (of a hypergraph).
 class EdgWalks
 
Syntaxc1wlks 39800 Extend class notation with 1-walks (of a hypergraph). TODO-AV: should be renamed to Walks after the current definition of Walks becomes obsolete.
 class 1Walks
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