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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.)
Vtx       iEdg              VtxDeg

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.)
Vtx Vtx       iEdg iEdg       VtxDeg VtxDeg

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.)
Vtx       Edg       UHGraph VtxDeg

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.)
iEdg       iEdg       Vtx       Vtx        Vtx                                    iEdg        VtxDeg VtxDegVtxDeg

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.)
iEdg       iEdg       Vtx       Vtx        Vtx                                    iEdg                      VtxDeg VtxDeg VtxDeg

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.)
iEdg       iEdg       Vtx       Vtx        Vtx               UHGraph        UHGraph               iEdg        VtxDeg VtxDegVtxDeg

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.)
iEdg       iEdg       Vtx       Vtx        Vtx               UHGraph        UHGraph               iEdg                      VtxDeg VtxDeg VtxDeg

Theoremvtxdlfgrval 39708* The value of the vertex degree function for a loop-free graph . (Contributed by AV, 23-Feb-2021.)
Vtx       iEdg              VtxDeg

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.)
Vtx       iEdg              VtxDeg       UMGraph

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.)
Vtx       iEdg              VtxDeg       USGraph

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.)
Vtx       iEdg       VtxDeg

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.
Vtx       Edg       USHGraph iEdg iEdg

Theoremvtxdushgrfvedg 39713* The value of the vertex degree function for a simple hypergraph. (Contributed by AV, 12-Dec-2020.)
Vtx       Edg       VtxDeg       USHGraph

Theoremvtxdusgrfvedg 39714* The value of the vertex degree function for a simple graph. (Contributed by AV, 12-Dec-2020.)
Vtx       Edg       VtxDeg       USGraph

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.)
Vtx       Edg       VtxDeg       UHGraph

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.)
Vtx       Edg       VtxDeg       UMGraph

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.)
Vtx       Edg       VtxDeg       UHGraph

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.)
Vtx       Edg       VtxDeg       USGraph

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.)
Vtx       Edg       VtxDeg       USGraph

Theoremvtxdgfusgrf 39720 The vertex degree function on finite simple graphs is a function from vertices to nonnegative integers. (Contributed by AV, 12-Dec-2020.)
Vtx       FinUSGraph VtxDeg

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.)
Vtx       FinUSGraph VtxDeg

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.)
Vtx                      iEdg        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.)
Vtx                      iEdg        Edg

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.)
Vtx                      iEdg        NeighbVtx

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.)
Vtx                      iEdg        VtxDeg

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.)
Vtx                      iEdg               VtxDeg

Theorem1hevtxdg0 39727 The vertex degree of vertex in a graph with only one hyperedge is 0 if is not incident with the edge . (Contributed by AV, 2-Mar-2021.)
iEdg        Vtx                                    VtxDeg

Theorem1hevtxdg1 39728 The vertex degree of vertex in a graph with only one hyperedge (not being a loop) is 1 if is incident with the edge . (Contributed by AV, 2-Mar-2021.)
iEdg        Vtx                                           VtxDeg

Theorem1hegrlfgr 39729* --- TODO-AV: not used anymore!? ! ------------------------- A graph with one hyperedge joining at least two vertices is a loop-free graph. (Contributed by AV, 23-Feb-2021.)
iEdg               iEdg

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.)
iEdg               Vtx        VtxDeg

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.)
iEdg               Vtx        VtxDeg

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.)
Vtx                                    iEdg        VtxDeg

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.)
Vtx                                    iEdg        VtxDeg

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.)
Vtx                                                  iEdg        VtxDeg

Theoremp1evtxdeqlem 39735 Lemma for p1evtxdeq 39736 and p1evtxdp1 39737. (Contributed by AV, 3-Mar-2021.)
Vtx       iEdg              Vtx        iEdg                                    VtxDeg VtxDegVtxDeg

Theoremp1evtxdeq 39736 If an edge which does not contain vertex is added to a graph (yielding a graph ), the degree of is the same in both graphs. (Contributed by AV, 2-Mar-2021.)
Vtx       iEdg              Vtx        iEdg                                           VtxDeg VtxDeg

Theoremp1evtxdp1 39737 If an edge (not being a loop) which contains vertex is added to a graph (yielding a graph ), the degree of is increased by 1. (Contributed by AV, 3-Mar-2021.)
Vtx       iEdg              Vtx        iEdg                                                  VtxDeg VtxDeg

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.)
Vtx

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.)
Vtx

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.)
iEdg

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.)
Edg

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.)
NeighbVtx

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.)
VtxDeg

Theoremumgr2v2evtx 39744 The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
Vtx

Theoremumgr2v2evtxel 39745 A vertex in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
Vtx

Theoremumgr2v2eiedg 39746 The edge function in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
iEdg

Theoremumgr2v2eedg 39747 The set of edges in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
Edg

Theoremumgr2v2e 39748 A multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)
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.)
NeighbVtx

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.)
VtxDeg

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.)
Vtx       USGraph NeighbVtx VtxDeg

Theoremusgruvtxvdb 39752 In a finite simple graph with n vertices a vertex is universal iff the vertex has degree . (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
Vtx       FinUSGraph UnivVtx VtxDeg

Theoremvdiscusgrb 39753* A finite simple graph with n vertices is complete iff every vertex has degree . (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 22-Dec-2020.)
Vtx       FinUSGraph ComplUSGraph VtxDeg

Theoremvdiscusgr 39754* In a finite complete simple graph with n vertices every vertex has degree . (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 17-Dec-2020.)
Vtx       FinUSGraph VtxDeg 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.)
Vtx       Edg       USGraph VtxDeg

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.)
Vtx       Edg       USGraph VtxDeg

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.)
Vtx       Edg       UHGraph VtxDeg

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.)
Vtx       Edg       USGraph VtxDeg

Theoremvdegp1ai-av 39759* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
Vtx              iEdg       Word        VtxDeg        Vtx                                    iEdg ++        VtxDeg

Theoremvdegp1bi-av 39760* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
Vtx              iEdg       Word        VtxDeg        Vtx                      iEdg ++        VtxDeg

Theoremvdegp1ci-av 39761* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.) (Revised by AV, 3-Mar-2021.)
Vtx              iEdg       Word        VtxDeg        Vtx                      iEdg ++        VtxDeg

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 NN0* VtxVtxDeg . This function, however, would not be defined for (see rgrx0nd 39798), because VtxVtxDeg is not a set (see rgrprcx 39796). It is expected that this function is not defined for every NN0* (how could this be proven?).

Syntaxcrgr 39762 Extend class notation to include the class of all regular graphs.
RegGraph

Syntaxcrusgr 39763 Extend class notation to include the class of all regular simple graphs.
RegUSGraph

Definitiondf-rgr 39764* Define the "k-regular" predicate, which is true for a "graph" being k-regular: read RegGraph as " is -regular" or " is a -regular graph". Note that is allowed to be positive infinity ( NN0*), as proposed by GL. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 26-Dec-2020.)
RegGraph NN0* VtxVtxDeg

Definitiondf-rusgr 39765* Define the "k-regular simple graph" predicate, which is true for a simple graph being k-regular: read RegUSGraph as is a -regular simple graph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 18-Dec-2020.)
RegUSGraph USGraph RegGraph

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.)
Vtx       VtxDeg       RegGraph NN0*

Theoremrgrprop 39767* The properties of a k-regular graph. (Contributed by Alexander van der Vekens, 8-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Vtx       VtxDeg       RegGraph NN0*

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.)
RegUSGraph USGraph RegGraph

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.)
RegUSGraph USGraph RegGraph

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.)
RegUSGraph RegGraph

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.)
RegUSGraph 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.)
Vtx       VtxDeg       RegUSGraph USGraph NN0*

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.)
Vtx       VtxDeg       RegUSGraph USGraph NN0*

Theoremusgreqdrusgr 39774* If all vertices in a simple graph have the same degree, the graph is k-regular. (Contributed by AV, 26-Dec-2020.)
Vtx       VtxDeg       USGraph NN0* RegUSGraph

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.)
Vtx       VtxDeg       FinUSGraph

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.)
Vtx       VtxDeg       FinUSGraph RegUSGraph

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.)
iEdg RegGraph

Theoremuhgr0edg0rgr 39778 A hypergraph is 0-regular if it has no edges. (Contributed by AV, 19-Dec-2020.)
UHGraph Edg RegGraph

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.)
UHGraph RegGraph Edg

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.)
USGraph RegUSGraph Edg

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.)
Vtx NN0* RegGraph

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.)
Vtx iEdg NN0* RegUSGraph

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.)
UHGraph Vtx NN0* RegUSGraph

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.)
NN0* RegUSGraph

Theorem0grrgr 39785 The null graph represented by an empty set is k-regular for every k. (Contributed by AV, 26-Dec-2020.)
NN0* RegGraph

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.)
Vtx       ComplUSGraph RegUSGraph

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 , then the assumption could be removed. (Contributed by Alexander van der Vekens, 14-Jul-2018.) (Revised by AV, 26-Dec-2020.)
Vtx       FinUSGraph ComplUSGraph RegUSGraph

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.)
Vtx       RegUSGraph USGraph NN0* NeighbVtx

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.)
Vtx       RegUSGraph USGraph NN0* Edg

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.)
Vtx       RegUSGraph USGraph NN0* Edg

Theoremrusgr1vtxlem 39791* Lemma for rusgr1vtx 39792. (Contributed by AV, 27-Dec-2020.)

Theoremrusgr1vtx 39792 If a k-regular simple graph has only one vertex, then k must be . (Contributed by Alexander van der Vekens, 4-Sep-2018.) (Revised by AV, 27-Dec-2020.)
Vtx RegUSGraph

Theoremrgrusgrprc 39793* The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
USGraph VtxVtxDeg

Theoremrusgrprc 39794 The class of 0-regular simple graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
RegUSGraph

Theoremrgrprc 39795 The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
RegGraph

Theoremrgrprcx 39796* The class of 0-regular graphs is a proper class. (Contributed by AV, 27-Dec-2020.)
VtxVtxDeg

Theoremrgrx0ndm 39797* 0 is not in the domain of the potentially alternatively defined vertex degree function. (Contributed by AV, 28-Dec-2020.)
NN0* VtxVtxDeg

Theoremrgrx0nd 39798* The potentially alternatively defined vertex degree function is not defined for 0. (Contributed by AV, 28-Dec-2020.)
NN0* VtxVtxDeg

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).
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.
1Walks

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