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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | finsumvtxdg2ssteplem3 28801* | Lemma for finsumvtxdg2sstep 28803. (Contributed by AV, 19-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (iEdgβπΊ) & β’ πΎ = (π β {π}) & β’ πΌ = {π β dom πΈ β£ π β (πΈβπ)} & β’ π = (πΈ βΎ πΌ) & β’ π = β¨πΎ, πβ© & β’ π½ = {π β dom πΈ β£ π β (πΈβπ)} β β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β (Ξ£π£ β (π β {π})(β―β{π β π½ β£ π£ β (πΈβπ)}) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}})) = (β―βπ½)) | ||
Theorem | finsumvtxdg2ssteplem4 28802* | Lemma for finsumvtxdg2sstep 28803. (Contributed by AV, 12-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (iEdgβπΊ) & β’ πΎ = (π β {π}) & β’ πΌ = {π β dom πΈ β£ π β (πΈβπ)} & β’ π = (πΈ βΎ πΌ) & β’ π = β¨πΎ, πβ© & β’ π½ = {π β dom πΈ β£ π β (πΈβπ)} β β’ ((((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β§ Ξ£π£ β πΎ ((VtxDegβπ)βπ£) = (2 Β· (β―βπ))) β (Ξ£π£ β (π β {π})((VtxDegβπΊ)βπ£) + ((β―βπ½) + (β―β{π β dom πΈ β£ (πΈβπ) = {π}}))) = (2 Β· ((β―βπ) + (β―βπ½)))) | ||
Theorem | finsumvtxdg2sstep 28803* | Induction step of finsumvtxdg2size 28804: In a finite pseudograph of finite size, the sum of the degrees of all vertices of the pseudograph is twice the size of the pseudograph if the sum of the degrees of all vertices of the subgraph of the pseudograph not containing one of the vertices is twice the size of the subgraph. (Contributed by AV, 19-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΈ = (iEdgβπΊ) & β’ πΎ = (π β {π}) & β’ πΌ = {π β dom πΈ β£ π β (πΈβπ)} & β’ π = (πΈ βΎ πΌ) & β’ π = β¨πΎ, πβ© β β’ (((πΊ β UPGraph β§ π β π) β§ (π β Fin β§ πΈ β Fin)) β ((π β Fin β Ξ£π£ β πΎ ((VtxDegβπ)βπ£) = (2 Β· (β―βπ))) β Ξ£π£ β π ((VtxDegβπΊ)βπ£) = (2 Β· (β―βπΈ)))) | ||
Theorem | finsumvtxdg2size 28804* |
The sum of the degrees of all vertices of a finite pseudograph of finite
size is twice the size of the pseudograph. See equation (1) in section
I.1 in [Bollobas] p. 4. Here, the
"proof" is simply the statement
"Since each edge has two endvertices, the sum of the degrees is
exactly
twice the number of edges". The formal proof of this theorem (for
pseudographs) is much more complicated, taking also the used auxiliary
theorems into account. The proof for a (finite) simple graph (see
fusgr1th 28805) would be shorter, but nevertheless still
laborious.
Although this theorem would hold also for infinite pseudographs and
pseudographs of infinite size, the proof of this most general version
(see theorem "sumvtxdg2size" below) would require many more
auxiliary
theorems (e.g., the extension of the sum Ξ£
over an arbitrary
set).
I dedicate this theorem and its proof to Norman Megill, who deceased too early on December 9, 2021. This proof is an example for the rigor which was the main motivation for Norman Megill to invent and develop Metamath, see section 1.1.6 "Rigor" on page 19 of the Metamath book: "... it is usually assumed in mathematical literature that the person reading the proof is a mathematician familiar with the specialty being described, and that the missing steps are obvious to such a reader or at least the reader is capable of filling them in." I filled in the missing steps of Bollobas' proof as Norm would have liked it... (Contributed by Alexander van der Vekens, 19-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β Ξ£π£ β π (π·βπ£) = (2 Β· (β―βπΌ))) | ||
Theorem | fusgr1th 28805* | The sum of the degrees of all vertices of a finite simple graph is twice the size of the graph. See equation (1) in section I.1 in [Bollobas] p. 4. Also known as the "First Theorem of Graph Theory" (see https://charlesreid1.com/wiki/First_Theorem_of_Graph_Theory). (Contributed by AV, 26-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ π· = (VtxDegβπΊ) β β’ (πΊ β FinUSGraph β Ξ£π£ β π (π·βπ£) = (2 Β· (β―βπΌ))) | ||
Theorem | finsumvtxdgeven 28806* | The sum of the degrees of all vertices of a finite pseudograph of finite size is even. See equation (2) in section I.1 in [Bollobas] p. 4, where it is also called the handshaking lemma. (Contributed by AV, 22-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β 2 β₯ Ξ£π£ β π (π·βπ£)) | ||
Theorem | vtxdgoddnumeven 28807* | The number of vertices of odd degree is even in a finite pseudograph of finite size. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 22-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β UPGraph β§ π β Fin β§ πΌ β Fin) β 2 β₯ (β―β{π£ β π β£ Β¬ 2 β₯ (π·βπ£)})) | ||
Theorem | fusgrvtxdgonume 28808* | The number of vertices of odd degree is even in a finite simple graph. Proposition 1.2.1 in [Diestel] p. 5. See also remark about equation (2) in section I.1 in [Bollobas] p. 4. (Contributed by AV, 27-Dec-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ π· = (VtxDegβπΊ) β β’ (πΊ β FinUSGraph β 2 β₯ (β―β{π£ β π β£ Β¬ 2 β₯ (π·βπ£)})) | ||
With df-rgr 28811 and df-rusgr 28812, 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 = (π β β0* β¦ {π β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = π}). This function, however, would not be defined at least for π = 0 (see rgrx0nd 28848), because {π β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = 0} is not a set (see rgrprcx 28846). It is expected that this function is not defined for every π β β0* (how could this be proven?). | ||
Syntax | crgr 28809 | Extend class notation to include the class of all regular graphs. |
class RegGraph | ||
Syntax | crusgr 28810 | Extend class notation to include the class of all regular simple graphs. |
class RegUSGraph | ||
Definition | df-rgr 28811* | Define the "k-regular" predicate, which is true for a "graph" being k-regular: read πΊ RegGraph πΎ as "πΊ is πΎ-regular" or "πΊ is a πΎ-regular graph". Note that πΎ is allowed to be positive infinity (πΎ β β0*), as proposed by GL. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
β’ RegGraph = {β¨π, πβ© β£ (π β β0* β§ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = π)} | ||
Definition | df-rusgr 28812* | Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read πΊ RegUSGraph πΎ as πΊ is a πΎ-regular simple graph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
β’ RegUSGraph = {β¨π, πβ© β£ (π β USGraph β§ π RegGraph π)} | ||
Theorem | isrgr 28813* | 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.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β π β§ πΎ β π) β (πΊ RegGraph πΎ β (πΎ β β0* β§ βπ£ β π (π·βπ£) = πΎ))) | ||
Theorem | rgrprop 28814* | The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ (πΊ RegGraph πΎ β (πΎ β β0* β§ βπ£ β π (π·βπ£) = πΎ)) | ||
Theorem | isrusgr 28815 | The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
β’ ((πΊ β π β§ πΎ β π) β (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΊ RegGraph πΎ))) | ||
Theorem | rusgrprop 28816 | The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΊ RegGraph πΎ)) | ||
Theorem | rusgrrgr 28817 | 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.) |
β’ (πΊ RegUSGraph πΎ β πΊ RegGraph πΎ) | ||
Theorem | rusgrusgr 28818 | A k-regular simple graph is a simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 18-Dec-2020.) |
β’ (πΊ RegUSGraph πΎ β πΊ β USGraph) | ||
Theorem | finrusgrfusgr 28819 | A finite regular simple graph is a finite simple graph. (Contributed by AV, 3-Jun-2021.) |
β’ π = (VtxβπΊ) β β’ ((πΊ RegUSGraph πΎ β§ π β Fin) β πΊ β FinUSGraph) | ||
Theorem | isrusgr0 28820* | The property of being a k-regular simple graph. (Contributed by Alexander van der Vekens, 7-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β π β§ πΎ β π) β (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (π·βπ£) = πΎ))) | ||
Theorem | rusgrprop0 28821* | The properties of a k-regular simple graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (π·βπ£) = πΎ)) | ||
Theorem | usgreqdrusgr 28822* | If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (π·βπ£) = πΎ) β πΊ RegUSGraph πΎ) | ||
Theorem | fusgrregdegfi 28823* | 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.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β FinUSGraph β§ π β β ) β (βπ£ β π (π·βπ£) = πΎ β πΎ β β0)) | ||
Theorem | fusgrn0eqdrusgr 28824* | 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.) |
β’ π = (VtxβπΊ) & β’ π· = (VtxDegβπΊ) β β’ ((πΊ β FinUSGraph β§ π β β ) β (βπ£ β π (π·βπ£) = πΎ β πΊ RegUSGraph πΎ)) | ||
Theorem | frusgrnn0 28825 | In a nonempty finite k-regular simple graph, the degree of each vertex is finite. (Contributed by AV, 7-May-2021.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β FinUSGraph β§ πΊ RegUSGraph πΎ β§ π β β ) β πΎ β β0) | ||
Theorem | 0edg0rgr 28826 | 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.) |
β’ ((πΊ β π β§ (iEdgβπΊ) = β ) β πΊ RegGraph 0) | ||
Theorem | uhgr0edg0rgr 28827 | A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.) |
β’ ((πΊ β UHGraph β§ (EdgβπΊ) = β ) β πΊ RegGraph 0) | ||
Theorem | uhgr0edg0rgrb 28828 | 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.) |
β’ (πΊ β UHGraph β (πΊ RegGraph 0 β (EdgβπΊ) = β )) | ||
Theorem | usgr0edg0rusgr 28829 | 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.) |
β’ (πΊ β USGraph β (πΊ RegUSGraph 0 β (EdgβπΊ) = β )) | ||
Theorem | 0vtxrgr 28830* | A null 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.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β ) β βπ β β0* πΊ RegGraph π) | ||
Theorem | 0vtxrusgr 28831* | 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.) |
β’ ((πΊ β π β§ (VtxβπΊ) = β β§ (iEdgβπΊ) = β ) β βπ β β0* πΊ RegUSGraph π) | ||
Theorem | 0uhgrrusgr 28832* | 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.) |
β’ ((πΊ β UHGraph β§ (VtxβπΊ) = β ) β βπ β β0* πΊ RegUSGraph π) | ||
Theorem | 0grrusgr 28833 | The null graph represented by an empty set is a k-regular simple graph for every k. (Contributed by AV, 26-Dec-2020.) |
β’ βπ β β0* β RegUSGraph π | ||
Theorem | 0grrgr 28834 | The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.) |
β’ βπ β β0* β RegGraph π | ||
Theorem | cusgrrusgr 28835 | 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.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β ComplUSGraph β§ π β Fin β§ π β β ) β πΊ RegUSGraph ((β―βπ) β 1)) | ||
Theorem | cusgrm1rusgr 28836 | A finite simple graph with n vertices is complete iff it is (n-1)-regular. Hint: If the definition of RegGraph was allowed for π β β€, then the assumption π β β could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.) |
β’ π = (VtxβπΊ) β β’ ((πΊ β FinUSGraph β§ π β β ) β (πΊ β ComplUSGraph β πΊ RegUSGraph ((β―βπ) β 1))) | ||
Theorem | rusgrpropnb 28837* | 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.) |
β’ π = (VtxβπΊ) β β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β(πΊ NeighbVtx π£)) = πΎ)) | ||
Theorem | rusgrpropedg 28838* | The properties of a k-regular simple graph expressed with edges. (Contributed by AV, 23-Dec-2020.) (Revised by AV, 27-Dec-2020.) |
β’ π = (VtxβπΊ) β β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β{π β (EdgβπΊ) β£ π£ β π}) = πΎ)) | ||
Theorem | rusgrpropadjvtx 28839* | 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.) |
β’ π = (VtxβπΊ) β β’ (πΊ RegUSGraph πΎ β (πΊ β USGraph β§ πΎ β β0* β§ βπ£ β π (β―β{π β π β£ {π£, π} β (EdgβπΊ)}) = πΎ)) | ||
Theorem | rusgrnumwrdl2 28840* | In a k-regular simple graph, the number of edges resp. walks of length 1 (represented as words of length 2) starting at a fixed vertex is k. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 6-May-2021.) |
β’ π = (VtxβπΊ) β β’ ((πΊ RegUSGraph πΎ β§ π β π) β (β―β{π€ β Word π β£ ((β―βπ€) = 2 β§ (π€β0) = π β§ {(π€β0), (π€β1)} β (EdgβπΊ))}) = πΎ) | ||
Theorem | rusgr1vtxlem 28841* | Lemma for rusgr1vtx 28842. (Contributed by AV, 27-Dec-2020.) |
β’ (((βπ£ β π (β―βπ΄) = πΎ β§ βπ£ β π π΄ = β ) β§ (π β π β§ (β―βπ) = 1)) β πΎ = 0) | ||
Theorem | rusgr1vtx 28842 | 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.) |
β’ (((β―β(VtxβπΊ)) = 1 β§ πΊ RegUSGraph πΎ) β πΎ = 0) | ||
Theorem | rgrusgrprc 28843* | The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
β’ {π β USGraph β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = 0} β V | ||
Theorem | rusgrprc 28844 | The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
β’ {π β£ π RegUSGraph 0} β V | ||
Theorem | rgrprc 28845 | The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
β’ {π β£ π RegGraph 0} β V | ||
Theorem | rgrprcx 28846* | The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.) |
β’ {π β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = 0} β V | ||
Theorem | rgrx0ndm 28847* | 0 is not in the domain of the potentially alternative definition of the sets of k-regular graphs for each extended nonnegative integer k. (Contributed by AV, 28-Dec-2020.) |
β’ π = (π β β0* β¦ {π β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = π}) β β’ 0 β dom π | ||
Theorem | rgrx0nd 28848* | The potentially alternatively defined k-regular graphs is not defined for k=0. (Contributed by AV, 28-Dec-2020.) |
β’ π = (π β β0* β¦ {π β£ βπ£ β (Vtxβπ)((VtxDegβπ)βπ£) = π}) β β’ (π β0) = β | ||
A "walk" in 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, or "A walk (of length k) in a graph is a nonempty alternating sequence v0 e0 v1 e1 ... e(k-1) vk of vertices and edges in G such that ei = { vi , vi+1 } for all i < k.", see definition of [Diestel] p. 10. Formalizing these definitions (mainly by representing the indexed vertices and edges by functions), a walk is represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges (e is a third function enumerating the edges within the graph, not within the walk), and p enumerates the vertices, see df-wlks 28853. Hence a walk (of length n) is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Alternatively, one could define a walk as a function π€:(0...(2 Β· π))βΆ((EdgβπΊ) βͺ (VtxβπΊ)) such that for all 0 β€ π β€ π, (π€β(2 Β· π)) β (VtxβπΊ) and for all 0 β€ π β€ (π β 1), (π€β((2 Β· π) + 1)) β (EdgβπΊ) and {(π€β(2 Β· π)), (π€β((2 Β· π) + 2))} β (π€β((2 Β· π) + 1)). Based on our definition of Walks, the class of all walks, more restrictive constructs are defined: * Trails (df-trls 28946): A "walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5, i.e., f(i) =/= f(j) if i =/= j. * Paths (df-pths 28970): A path is a walk whose vertices except the first and the last vertex are distinct, i.e., p(i) =/= p(j) if i < j, except possibly when i = 0 and j = n. * SPaths (simple paths, df-spths 28971): A simple path "is a walk with distinct vertices.", see Notation of [Bollobas] p. 5, i.e., p(i) =/= p(j) if i =/= j. * ClWalks (closed walks, df-clwlks 29025): A walk whose endvertices coincide is called a closed walk, i.e., p(0) = p(n). * Circuits (df-crcts 29040): "A trail whose endvertices coincide (a closed trail) is called a circuit." (see Definition of [Bollobas] p. 5), i.e., f(i) =/= f(j) if i =/= j and p(0) = p(n). Equivalently, a circuit is a closed walk with distinct edges. * Cycles (df-cycls 29041): A path whose endvertices coincide (a closed path) is called a cycle, i.e., p(i) =/= p(j) if i =/= j, except i = 0 and j = n, and p(0) = p(n). Equivalently, a cycle is a closed walk with distinct vertices. * EulerPaths (Eulerian paths, df-eupth 29448): An Eulerian path is "a trail containing all edges [of the graph]" (see definition in [Bollobas] p. 16), i.e., f(i) =/= f(j) if i =/= j and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Note, however, that an Eulerian path needs not be a path. * Eulerian circuit: An Eulerian circuit (called Euler tour in the definition in [Diestel] p. 22) is "a circuit in a graph containing all the edges" (see definition in [Bollobas] p. 16), i.e., f(i) =/= f(j) if i =/= j, p(0) = p(n) and for all edges e(x) there is an 1 <= i <= n with e(x) = e(f(i)). Hierarchy of all kinds of walks (apply ssriv 3986 and elopabran 5562 to the mentioned theorems to obtain the following subset relationships, as available for clwlkiswlk 29028, see clwlkwlk 29029 and clwlkswks 29030): * Trails are walks (trliswlk 28951): (TrailsβπΊ) β (WalksβπΊ) * Paths are trails (pthistrl 28979): (PathsβπΊ) β (TrailsβπΊ) * Simple paths are paths (spthispth 28980): (SPathsβπΊ) β (PathsβπΊ) * Closed walks are walks (clwlkiswlk 29028): (ClWalksβπΊ) β (WalksβπΊ) * Circuits are closed walks (crctisclwlk 29048): (CircuitsβπΊ) β (ClWalksβπΊ) * Circuits are trails (crctistrl 29049): (CircuitsβπΊ) β (TrailsβπΊ) * Cycles are paths (cyclispth 29051): (CyclesβπΊ) β (PathsβπΊ) * Cycles are circuits (cycliscrct 29053): (CyclesβπΊ) β (CircuitsβπΊ) * (Non-trivial) cycles are not simple paths (cyclnspth 29054): (πΉ β β β (πΉ(CyclesβπΊ)π β Β¬ πΉ(SPathsβπΊ)π)) * Eulerian paths are trails (eupthistrl 29461): (EulerPathsβπΊ) β (TrailsβπΊ) Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e., omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 14464, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definition df-wwlks 29081 for WWalks, and the representation of a walk as sequence of its vertices is called "walk as word". Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In this case, the general definitions of walks and the definition of walks as words are equivalent, see wlkiswwlks 29127. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph). Based on this definition of WWalks, the class of all walks as word, more restrictive constructs are defined analogously to the general definition of a walk: * WWalksN (walks of length N as word, df-wwlksn 29082): n = N * WSPathsN (simple paths of length N as word, df-wspthsn 29084): p(i) =/= p(j) if i =/= j and n = N * ClWWalks (closed walks as word, df-clwwlk 29232): p(0) = p(n) * ClWWalksN (closed walks of length N as word, df-clwwlkn 29275): p(0) = p(n) and n = N Finally, there are a couple of definitions for (special) walks β¨πΉ, πβ© having fixed endpoints π΄ and π΅: * Walks with particular endpoints (df-wlkson 28854): πΉ(π΄(WalksOnβπΊ)π΅)π * Trails with particular endpoints (df-trlson 28947): πΉ(π΄(TrailsOnβπΊ)π΅)π * Paths with particular endpoints (df-pthson 28972): πΉ(π΄(PathsOnβπΊ)π΅)π * Simple paths with particular endpoints (df-spthson 28973): πΉ(π΄(SPathsOnβπΊ)π΅)π * Walks of a fixed length π as words with particular endpoints (df-wwlksnon 29083): (π΄(π WWalksNOn πΊ)π΅) * Simple paths of a fixed length π as words with particular endpoints (df-wspthsnon 29085): (π΄(π WSPathsNOn πΊ)π΅) * Closed Walks of a fixed length π as words anchored at a particular vertex π΄ (df-wwlksnon 29083): (π΄(ClWWalksNOnβπΊ)π) | ||
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 the 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 28852. 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 Walks, see df-wlks 28853, restricting s to 1. wlk1ewlk 28894 shows that such a 1-walk "on the vertex level" induces a 1-walk "on the edge level". | ||
Syntax | cewlks 28849 | Extend class notation with s-walks "on the edge level" (of a hypergraph). |
class EdgWalks | ||
Syntax | cwlks 28850 | Extend class notation with walks (i.e. 1-walks) (of a hypergraph). |
class Walks | ||
Syntax | cwlkson 28851 | Extend class notation with walks between two vertices (within a graph). |
class WalksOn | ||
Definition | df-ewlks 28852* | Define the set of all s-walks of edges (in a hypergraph) corresponding to s-walks "on the edge level" discussed in Aksoy et al. For an extended nonnegative integer s, an s-walk is a sequence of hyperedges, e(0), e(1), ... , e(k), where e(j-1) and e(j) have at least s vertices in common (for j=1, ... , k). In contrast to the definition in Aksoy et al., π = 0 (a 0-walk is an arbitrary sequence of hyperedges) and π = +β (then the number of common vertices of two adjacent hyperedges must be infinite) are allowed. Furthermore, it is not forbidden that adjacent hyperedges are equal. (Contributed by AV, 4-Jan-2021.) |
β’ EdgWalks = (π β V, π β β0* β¦ {π β£ [(iEdgβπ) / π](π β Word dom π β§ βπ β (1..^(β―βπ))π β€ (β―β((πβ(πβ(π β 1))) β© (πβ(πβπ)))))}) | ||
Definition | df-wlks 28853* |
Define the set of all walks (in a hypergraph). Such walks correspond to
the s-walks "on the vertex level" (with s = 1), and also to
1-walks "on
the edge level" (see wlk1walk 28893) discussed in Aksoy et al. The
predicate πΉ(WalksβπΊ)π can be read as "The pair
β¨πΉ, πβ© represents a walk in a graph
πΊ", see also iswlk 28864.
The condition {(πβπ), (πβ(π + 1))} β ((iEdgβπ)β(πβπ)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (πβπ) = (πβ(π + 1)) should be allowed only if there is a loop at (πβπ). Otherwise, C would be fulfilled by each edge containing (πβπ). According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the 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). (Contributed by AV, 30-Dec-2020.) |
β’ Walks = (π β V β¦ {β¨π, πβ© β£ (π β Word dom (iEdgβπ) β§ π:(0...(β―βπ))βΆ(Vtxβπ) β§ βπ β (0..^(β―βπ))if-((πβπ) = (πβ(π + 1)), ((iEdgβπ)β(πβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β ((iEdgβπ)β(πβπ))))}) | ||
Definition | df-wlkson 28854* | Define the collection of walks with particular endpoints (in a hypergraph). The predicate πΉ(π΄(WalksOnβπΊ)π΅)π can be read as "The pair β¨πΉ, πβ© represents a walk from vertex π΄ to vertex π΅ in a graph πΊ", see also iswlkon 28911. This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.) |
β’ WalksOn = (π β V β¦ (π β (Vtxβπ), π β (Vtxβπ) β¦ {β¨π, πβ© β£ (π(Walksβπ)π β§ (πβ0) = π β§ (πβ(β―βπ)) = π)})) | ||
Theorem | ewlksfval 28855* | The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β π β§ π β β0*) β (πΊ EdgWalks π) = {π β£ (π β Word dom πΌ β§ βπ β (1..^(β―βπ))π β€ (β―β((πΌβ(πβ(π β 1))) β© (πΌβ(πβπ)))))}) | ||
Theorem | isewlk 28856* | Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β π β§ π β β0* β§ πΉ β π) β (πΉ β (πΊ EdgWalks π) β (πΉ β Word dom πΌ β§ βπ β (1..^(β―βπΉ))π β€ (β―β((πΌβ(πΉβ(π β 1))) β© (πΌβ(πΉβπ))))))) | ||
Theorem | ewlkprop 28857* | Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ β (πΊ EdgWalks π) β ((πΊ β V β§ π β β0*) β§ πΉ β Word dom πΌ β§ βπ β (1..^(β―βπΉ))π β€ (β―β((πΌβ(πΉβ(π β 1))) β© (πΌβ(πΉβπ)))))) | ||
Theorem | ewlkinedg 28858 | The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΉ β (πΊ EdgWalks π) β§ πΎ β (1..^(β―βπΉ))) β π β€ (β―β((πΌβ(πΉβ(πΎ β 1))) β© (πΌβ(πΉβπΎ))))) | ||
Theorem | ewlkle 28859 | An s-walk of edges is also a t-walk of edges if π‘ β€ π . (Contributed by AV, 4-Jan-2021.) |
β’ ((πΉ β (πΊ EdgWalks π) β§ π β β0* β§ π β€ π) β πΉ β (πΊ EdgWalks π)) | ||
Theorem | upgrewlkle2 28860 | In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.) |
β’ ((πΊ β UPGraph β§ πΉ β (πΊ EdgWalks π) β§ 1 < (β―βπΉ)) β π β€ 2) | ||
Theorem | wkslem1 28861 | Lemma 1 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
β’ (π΄ = π΅ β (if-((πβπ΄) = (πβ(π΄ + 1)), (πΌβ(πΉβπ΄)) = {(πβπ΄)}, {(πβπ΄), (πβ(π΄ + 1))} β (πΌβ(πΉβπ΄))) β if-((πβπ΅) = (πβ(π΅ + 1)), (πΌβ(πΉβπ΅)) = {(πβπ΅)}, {(πβπ΅), (πβ(π΅ + 1))} β (πΌβ(πΉβπ΅))))) | ||
Theorem | wkslem2 28862 | Lemma 2 for walks to substitute the index of the condition for vertices and edges in a walk. (Contributed by AV, 23-Apr-2021.) |
β’ ((π΄ = π΅ β§ (π΄ + 1) = πΆ) β (if-((πβπ΄) = (πβ(π΄ + 1)), (πΌβ(πΉβπ΄)) = {(πβπ΄)}, {(πβπ΄), (πβ(π΄ + 1))} β (πΌβ(πΉβπ΄))) β if-((πβπ΅) = (πβπΆ), (πΌβ(πΉβπ΅)) = {(πβπ΅)}, {(πβπ΅), (πβπΆ)} β (πΌβ(πΉβπ΅))))) | ||
Theorem | wksfval 28863* | The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β π β (WalksβπΊ) = {β¨π, πβ© β£ (π β Word dom πΌ β§ π:(0...(β―βπ))βΆπ β§ βπ β (0..^(β―βπ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πβπ))))}) | ||
Theorem | iswlk 28864* | Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β π β§ πΉ β π β§ π β π) β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) | ||
Theorem | wlkprop 28865* | Properties of a walk. (Contributed by AV, 5-Nov-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))))) | ||
Theorem | wlkv 28866 | The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.) |
β’ (πΉ(WalksβπΊ)π β (πΊ β V β§ πΉ β V β§ π β V)) | ||
Theorem | iswlkg 28867* | Generalization of iswlk 28864: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β π β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) | ||
Theorem | wlkf 28868 | The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(WalksβπΊ)π β πΉ β Word dom πΌ) | ||
Theorem | wlkcl 28869 | A walk has length β―(πΉ), which is an integer. Formerly proven for an Eulerian path, see eupthcl 29460. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
β’ (πΉ(WalksβπΊ)π β (β―βπΉ) β β0) | ||
Theorem | wlkp 28870 | The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.) |
β’ π = (VtxβπΊ) β β’ (πΉ(WalksβπΊ)π β π:(0...(β―βπΉ))βΆπ) | ||
Theorem | wlkpwrd 28871 | The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.) |
β’ π = (VtxβπΊ) β β’ (πΉ(WalksβπΊ)π β π β Word π) | ||
Theorem | wlklenvp1 28872 | The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.) |
β’ (πΉ(WalksβπΊ)π β (β―βπ) = ((β―βπΉ) + 1)) | ||
Theorem | wksv 28873* | The class of walks is a set. (Contributed by AV, 15-Jan-2021.) (Proof shortened by SN, 11-Dec-2024.) |
β’ {β¨π, πβ© β£ π(WalksβπΊ)π} β V | ||
Theorem | wksvOLD 28874* | Obsolete version of wksv 28873 as of 11-Dec-2024. (Contributed by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ {β¨π, πβ© β£ π(WalksβπΊ)π} β V | ||
Theorem | wlkn0 28875 | The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
β’ (πΉ(WalksβπΊ)π β π β β ) | ||
Theorem | wlklenvm1 28876 | The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
β’ (πΉ(WalksβπΊ)π β (β―βπΉ) = ((β―βπ) β 1)) | ||
Theorem | ifpsnprss 28877 | Lemma for wlkvtxeledg 28878: Two adjacent (not necessarily different) vertices π΄ and π΅ in a walk are incident with an edge πΈ. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.) |
β’ (if-(π΄ = π΅, πΈ = {π΄}, {π΄, π΅} β πΈ) β {π΄, π΅} β πΈ) | ||
Theorem | wlkvtxeledg 28878* | Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(WalksβπΊ)π β βπ β (0..^(β―βπΉ)){(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))) | ||
Theorem | wlkvtxiedg 28879* | The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(WalksβπΊ)π β βπ β (0..^(β―βπΉ))βπ β ran πΌ{(πβπ), (πβ(π + 1))} β π) | ||
Theorem | relwlk 28880 | The set (WalksβπΊ) of all walks on πΊ is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.) |
β’ Rel (WalksβπΊ) | ||
Theorem | wlkvv 28881 | If there is at least one walk in the graph, all walks are in the universal class of ordered pairs. (Contributed by AV, 2-Jan-2021.) |
β’ ((1st βπ)(WalksβπΊ)(2nd βπ) β π β (V Γ V)) | ||
Theorem | wlkop 28882 | A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
β’ (π β (WalksβπΊ) β π = β¨(1st βπ), (2nd βπ)β©) | ||
Theorem | wlkcpr 28883 | A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
β’ (π β (WalksβπΊ) β (1st βπ)(WalksβπΊ)(2nd βπ)) | ||
Theorem | wlk2f 28884* | If there is a walk π there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.) |
β’ (π β (WalksβπΊ) β βπβπ π(WalksβπΊ)π) | ||
Theorem | wlkcomp 28885* | A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ πΉ = (1st βπ) & β’ π = (2nd βπ) β β’ ((πΊ β π β§ π β (π Γ π)) β (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ)))))) | ||
Theorem | wlkcompim 28886* | Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ πΉ = (1st βπ) & β’ π = (2nd βπ) β β’ (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))if-((πβπ) = (πβ(π + 1)), (πΌβ(πΉβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β (πΌβ(πΉβπ))))) | ||
Theorem | wlkelwrd 28887 | The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ πΉ = (1st βπ) & β’ π = (2nd βπ) β β’ (π β (WalksβπΊ) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ)) | ||
Theorem | wlkeq 28888* | Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.) |
β’ ((π΄ β (WalksβπΊ) β§ π΅ β (WalksβπΊ) β§ π = (β―β(1st βπ΄))) β (π΄ = π΅ β (π = (β―β(1st βπ΅)) β§ βπ₯ β (0..^π)((1st βπ΄)βπ₯) = ((1st βπ΅)βπ₯) β§ βπ₯ β (0...π)((2nd βπ΄)βπ₯) = ((2nd βπ΅)βπ₯)))) | ||
Theorem | edginwlk 28889 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.) |
β’ πΌ = (iEdgβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((Fun πΌ β§ πΉ β Word dom πΌ β§ πΎ β (0..^(β―βπΉ))) β (πΌβ(πΉβπΎ)) β πΈ) | ||
Theorem | upgredginwlk 28890 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) |
β’ πΌ = (iEdgβπΊ) & β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ β Word dom πΌ) β (πΎ β (0..^(β―βπΉ)) β (πΌβ(πΉβπΎ)) β πΈ)) | ||
Theorem | iedginwlk 28891 | The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((Fun πΌ β§ πΉ(WalksβπΊ)π β§ π β (0..^(β―βπΉ))) β (πΌβ(πΉβπ)) β ran πΌ) | ||
Theorem | wlkl1loop 28892 | A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.) |
β’ (((Fun (iEdgβπΊ) β§ πΉ(WalksβπΊ)π) β§ ((β―βπΉ) = 1 β§ (πβ0) = (πβ1))) β {(πβ0)} β (EdgβπΊ)) | ||
Theorem | wlk1walk 28893* | A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.) |
β’ πΌ = (iEdgβπΊ) β β’ (πΉ(WalksβπΊ)π β βπ β (1..^(β―βπΉ))1 β€ (β―β((πΌβ(πΉβ(π β 1))) β© (πΌβ(πΉβπ))))) | ||
Theorem | wlk1ewlk 28894 | A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.) |
β’ (πΉ(WalksβπΊ)π β πΉ β (πΊ EdgWalks 1)) | ||
Theorem | upgriswlk 28895* | Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) β β’ (πΊ β UPGraph β (πΉ(WalksβπΊ)π β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}))) | ||
Theorem | upgrwlkedg 28896* | The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.) |
β’ πΌ = (iEdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(WalksβπΊ)π) β βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))}) | ||
Theorem | upgrwlkcompim 28897* | Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.) |
β’ π = (VtxβπΊ) & β’ πΌ = (iEdgβπΊ) & β’ πΉ = (1st βπ) & β’ π = (2nd βπ) β β’ ((πΊ β UPGraph β§ π β (WalksβπΊ)) β (πΉ β Word dom πΌ β§ π:(0...(β―βπΉ))βΆπ β§ βπ β (0..^(β―βπΉ))(πΌβ(πΉβπ)) = {(πβπ), (πβ(π + 1))})) | ||
Theorem | wlkvtxedg 28898* | The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
β’ πΈ = (EdgβπΊ) β β’ (πΉ(WalksβπΊ)π β βπ β (0..^(β―βπΉ))βπ β πΈ {(πβπ), (πβ(π + 1))} β π) | ||
Theorem | upgrwlkvtxedg 28899* | The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) |
β’ πΈ = (EdgβπΊ) β β’ ((πΊ β UPGraph β§ πΉ(WalksβπΊ)π) β βπ β (0..^(β―βπΉ)){(πβπ), (πβ(π + 1))} β πΈ) | ||
Theorem | uspgr2wlkeq 28900* | Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.) |
β’ ((πΊ β USPGraph β§ (π΄ β (WalksβπΊ) β§ π΅ β (WalksβπΊ)) β§ π = (β―β(1st βπ΄))) β (π΄ = π΅ β (π = (β―β(1st βπ΅)) β§ βπ¦ β (0...π)((2nd βπ΄)βπ¦) = ((2nd βπ΅)βπ¦)))) |
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