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Definition df-clwlk 30412
Description: Define the set of all Closed Walks (in an undirected graph).

According to Huneke: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0.

According to the definition of a walk as two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 30425! (Contributed by Alexander van der Vekens, 12-Mar-2018.)

Assertion
Ref Expression
df-clwlk  |- ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
Distinct variable group:    v, e, f, p

Detailed syntax breakdown of Definition df-clwlk
StepHypRef Expression
1 cclwlk 30409 . 2  class ClWalks
2 vv . . 3  setvar  v
3 ve . . 3  setvar  e
4 cvv 2970 . . 3  class  _V
5 vf . . . . . . 7  setvar  f
65cv 1368 . . . . . 6  class  f
7 vp . . . . . . 7  setvar  p
87cv 1368 . . . . . 6  class  p
92cv 1368 . . . . . . 7  class  v
103cv 1368 . . . . . . 7  class  e
11 cwalk 23403 . . . . . . 7  class Walks
129, 10, 11co 6089 . . . . . 6  class  ( v Walks 
e )
136, 8, 12wbr 4290 . . . . 5  wff  f ( v Walks  e ) p
14 cc0 9280 . . . . . . 7  class  0
1514, 8cfv 5416 . . . . . 6  class  ( p `
 0 )
16 chash 12101 . . . . . . . 8  class  #
176, 16cfv 5416 . . . . . . 7  class  ( # `  f )
1817, 8cfv 5416 . . . . . 6  class  ( p `
 ( # `  f
) )
1915, 18wceq 1369 . . . . 5  wff  ( p `
 0 )  =  ( p `  ( # `
 f ) )
2013, 19wa 369 . . . 4  wff  ( f ( v Walks  e ) p  /\  ( p `
 0 )  =  ( p `  ( # `
 f ) ) )
2120, 5, 7copab 4347 . . 3  class  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) }
222, 3, 4, 4, 21cmpt2 6091 . 2  class  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( f ( v Walks  e
) p  /\  (
p `  0 )  =  ( p `  ( # `  f ) ) ) } )
231, 22wceq 1369 1  wff ClWalks  =  ( v  e.  _V , 
e  e.  _V  |->  {
<. f ,  p >.  |  ( f ( v Walks 
e ) p  /\  ( p `  0
)  =  ( p `
 ( # `  f
) ) ) } )
Colors of variables: wff setvar class
This definition is referenced by:  clwlk  30415  isclwlkg  30417  clwlkiswlk  30419  clwlkswlks  30420  clwlkcompim  30424
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