LabeledGraphs
A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.
Simple undirected graphs on the labeled vertex set [n].
LabeledGraphs(...)Simple undirected graphs on the labeled vertex set [n].Typed signatures
| Type | Package | Applies |
|---|---|---|
(integer<0..>) -> indexed_collection<labeled_graph> | combinatorics | always |
Definition
Epsil, as the implementation runs it.
count(_n)
2 ^ (Binomial(_n, 2) if 0 <= 2 && 2 <= _n else 0)unrank(_n, _r)
Map(
Function(Apply((gu_k: integer) => Apply(
(gu_u: integer) => Apply(
(gu_v: integer) => [gu_v, gu_u] if False else [gu_u, gu_v],
gu_k -
((Binomial(_n, 2) if 0 <= 2 &&
2 <= _n else 0) -
(Binomial(_n - gu_u +
1, 2) if 0 <= 2 &&
2 <=
_n - gu_u +
1 else 0)) + gu_u,
),
_tables[gu_k],
), gk), gk),
Filter(
Range(1, Binomial(_n, 2) if 0 <= 2 && 2 <= _n else 0, 1),
Function((_r - _r % 2 ^ (gb - 1)) / 2 ^ (gb - 1) % 2 == 1, gb),
),
)rank(_n, _x)
Fold(Function(
ga + 2 ^ ((Binomial(_n, 2) if 0 <= 2 && 2 <= _n else 0) -
(Binomial(_n - Min(_x[gj][1], _x[gj][2]) + 1, 2) if 0 <=
2 &&
2 <=
_n -
Min(_x[gj][1], _x[gj][2]) +
1 else 0) + (Max(_x[gj][1], _x[gj][2]) - Min(
_x[gj][1],
_x[gj][2],
)) - 1),
ga,
gj,
), 0, Range(1, Length(_x), 1))valid(_n, _x)
Fold(Function(ok_wb && 1 <= _x[wb][1] && _x[wb][1] <= _n && 1 <= _x[wb][2] &&
_x[wb][2] <= _n && _x[wb][1] != _x[wb][2], ok_wb, wb), True, Range(1, Length(_x), 1)) && Fold(
Function(ok_wj &&
Fold(
Function(ok_wi &&
(Binomial(_n, 2) if 0 <= 2 &&
2 <= _n else 0) -
(Binomial(_n -
Min(_x[wi][1], _x[wi][2]) +
1, 2) if 0 <= 2 &&
2 <=
_n -
Min(_x[wi][1], _x[wi][2]) +
1 else 0) +
(Max(_x[wi][1], _x[wi][2]) -
Min(_x[wi][1], _x[wi][2])) !=
(Binomial(_n, 2) if 0 <= 2 &&
2 <= _n else 0) -
(Binomial(_n -
Min(_x[wj][1], _x[wj][2]) +
1, 2) if 0 <= 2 &&
2 <=
_n -
Min(_x[wj][1], _x[wj][2]) +
1 else 0) +
(Max(_x[wj][1], _x[wj][2]) -
Min(_x[wj][1], _x[wj][2])), ok_wi, wi),
True,
Range(1, wj - 1, 1),
), ok_wj, wj),
True,
Range(1, Length(_x), 1),
) if Fold(
Function(ok_wp &&
Length(_x[wp]) == 2, ok_wp, wp),
True,
Range(1, Length(_x), 1),
) else Falsetables(_n)
Fold(
Function(Join(et, Map(Function(eu, ez), Range(1, _n - eu, 1))), et, eu),
[],
Range(1, _n - 1, 1),
)Statistics
On LabeledGraph, called as CombinatorialStat(x, "Name").
- AlbertsonIndexSt001351The sum over the edges of the difference of the degrees of its ends.
- BridgeCountSt001479The edges whose deletion leaves more components.
- ChromaticNumberSt000098The fewest colors for the vertices so that adjacent vertices differ.
- CliqueNumberSt000097The most vertices that are pairwise adjacent.
- ComplementComponentCountSt000286The connected components of the complement graph.
- ConnectedInducedSubgraphCountSt001441The nonempty sets of vertices that induce a connected subgraph.
- CutVertexCountSt000552The vertices whose deletion leaves more components.
- CyclomaticNumberSt001311The edges less the vertices plus the components: the independent cycles.
- DegreeTwoVertexCountSt000313The vertices with exactly two edges.
- DistanceFourPairCountSt000449The pairs of vertices at distance 4.
- DistanceThreePairCountSt000447The pairs of vertices at distance 3.
- DistanceTwoPairCountSt000448The pairs of vertices at distance 2.
- EvenDegreeVertexCountSt000309The vertices with an even number of edges.
- FirstZagrebIndexSt000465The sum of the squares of the degrees.
- ForgottenIndexSt000571The sum of the cubes of the degrees.
- FourCliqueCountSt001309The sets of four vertices that are pairwise adjacent.
- GraphComponentCountSt000287The connected components: the vertices that are the first of their component.
- GraphEdgeCountSt000081The edges of the graph.
- GraphLeafCountSt000312The vertices with exactly one edge.
- HosoyaIndexSt000468The sets of edges with no two sharing a vertex, the empty set included.
- IndependenceNumberSt000093The most vertices that are pairwise non-adjacent.
- IndependentSetCountSt000300The sets of vertices, the empty set included, with no two adjacent.
- InducedPathOnThreeVerticesCountSt001308The sets of three vertices that induce a path: exactly two of the three edges.
- IsolatedVertexCountSt000315The vertices with no edge.
- MatchingNumberSt000387The most edges with no two sharing a vertex.
- MaximalIndependentSetCountSt001304The independent sets of vertices that no vertex can be added to.
- MaximumDegreeSt000171The greatest number of edges at one vertex.
- MaximumDegreeVertexCountSt001368The vertices whose degree is the greatest.
- MaximumDistanceSt000259The greatest distance between two vertices: the diameter.
- MinimumDegreeSt000310The fewest edges at one vertex.
- MinimumEccentricitySt000260The least eccentricity of a vertex, the greatest distance from it to another vertex: the radius.
- OddDegreeVertexCountSt000311The vertices with an odd number of edges.
- OreDegreeSt000915The greatest sum of the degrees of the two ends of an edge.
- SecondZagrebIndexSt000361The sum over the edges of the product of the degrees of its ends.
- TotalIrregularitySt001522The sum over the unordered pairs of vertices of the difference of their degrees.
- TriangleCountSt000095The triples of vertices that are pairwise adjacent.
- TriangleVertexCountSt001995The vertices on a triangle.
- VertexCoverNumberSt000362The fewest vertices that meet every edge: the vertices less the independence number.
- WienerIndexSt000265The sum of the distances over the pairs of vertices that are joined by a path.
Maps
Into and out of LabeledGraph, called as CombinatorialMap(x, "Name"). Properties are the laws we state and what FindStat records.
28 more in FindStat, not implemented here
- Mp00011BinaryTree → LabeledGraphGraded mapto graph
- Mp00037LabeledGraph → IntegerPartitionSurjectionGraded mapto partition of connected components
- Mp00046OrderedTree → LabeledGraphGraded mapto graph
- Mp00074Posets → LabeledGraphGraded mapto graph
- Mp00111LabeledGraph → LabeledGraphBijectionGraded mapInvolutioncomplement
- Mp00117LabeledGraph → LabeledGraphIdempotentGraded mapOre closure
- Mp00147LabeledGraph → LabeledGraphGraded mapsquare
- Mp00152LabeledGraph → IntegerCompositionSurjectionGraded mapLaplacian multiplicities
- Mp00154LabeledGraph → LabeledGraphIdempotentcore
- Mp00156LabeledGraph → LabeledGraphline graph
- Mp00157LabeledGraph → LabeledGraphGraded mapInvolutionconnected complement
- Mp00160Permutation → LabeledGraphGraded mapgraph of inversions
- Mp00184IntegerComposition → LabeledGraphInjectionGraded mapto threshold graph
- Mp00198Posets → LabeledGraphGraded mapincomparability graph
- Mp00203LabeledGraph → LabeledGraphInjectionGraded mapcone
- Mp00243LabeledGraph → Posetsweak duplicate order
- Mp00247LabeledGraph → LabeledGraphIdempotentde-duplicate
- Mp00250LabeledGraph → LabeledGraphclique graph
- Mp00251LabeledGraph → IntegerPartitionclique sizes
- Mp00259LabeledGraph → LabeledGraphInjectionGraded mapvertex addition
- Mp00264LabeledGraph → LabeledGraphIdempotentdelete endpoints
- Mp00266LabeledGraph → Latticesconnected vertex partitions
- Mp00274LabeledGraph → LabeledGraphblock-cut tree
- Mp00275LabeledGraph → IntegerPartitionSurjectionto edge-partition of connected components
- Mp00276LabeledGraph → IntegerPartitionto edge-partition of biconnected components
- Mp00318LabeledGraph → LabeledGraphInvolutiondual on components
- Mp00324LabeledGraph → IntegerCompositionGraded mapchromatic difference sequence
- Mp00335Lattices → LabeledGraphzero-divisor graph