LabeledGraph

A name from the enumeratio catalogue with no kernel of its own yet; this page carries its crosswalk only.

As a single-graded labeled_graph

LabeledGraph(...)As a single-graded labeled_graph

Typed signatures

Type	Package	Applies
(tuple<integer, list<list<integer>>>) -> labeled_graph	combinatorics	always

Domain: Combinatorics

Statistics

On LabeledGraph, called as CombinatorialStat(x, "Name").

• AboveAverageDegreeVertexCountSt001692The vertices whose degree is greater than the average degree.
• AbsoluteAdjacencyDeterminantSt001702The absolute value of the determinant of the adjacency matrix.
• AdjacencyDeterminantSt000351The determinant of the adjacency matrix.
• AlbertsonIndexSt001351The sum over the edges of the difference of the degrees of its ends.
• AnnihilationNumberSt001286The most vertices whose degrees add up to at most the number of edges.
• BipartiteEdgeDeletionNumberSt001572The fewest edges whose deletion leaves a bipartite graph.
• BlockCountSt000553The blocks: the maximal connected subgraphs with no cut vertex; an isolated vertex is a block.
• BridgeCountSt001479The edges whose deletion leaves more components.
• CelebrityCountSt001689The vertices of whom more than half the neighbors have a smaller degree.
• CenterEdgeCountSt001341The edges with both ends in the center.
• CenterVertexCountSt001342The vertices whose greatest distance to another vertex is the least.
• ChromaticCliqueGapSt001793The chromatic number less the clique number.
• ChromaticNumberSt000098The fewest colors for the vertices so that adjacent vertices differ.
• CliqueCocliqueNumberSt000926The clique number times the independence number.
• 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.
• DegeneracySt001277The greatest, over the sets of vertices, of the least number of neighbors a vertex has inside the set.
• DegreeRangeSt001742The greatest degree less the least.
• DegreeSumSt000350The sum of the degrees: twice the number of edges.
• DegreeTwoVertexCountSt000313The vertices with exactly two edges.
• DifferentDegreePairCountSt001708The pairs of vertices whose degrees differ.
• DissociationNumberSt001315The most vertices in a set in which every vertex has at most one neighbor.
• DistanceFourPairCountSt000449The pairs of vertices at distance 4.
• DistanceThreePairCountSt000447The pairs of vertices at distance 3.
• DistanceTwoPairCountSt000448The pairs of vertices at distance 2.
• DistinctNeighborhoodCountSt000722The different sets of neighbors the vertices have.
• DominatingSetCountSt001303The sets of vertices such that every vertex is in the set or adjacent to one of it.
• DominationNumberSt000273The fewest vertices such that every vertex is one of them or adjacent to one.
• EulerFaceCountSt000450The edges less the vertices plus 2: the faces of a connected planar graph.
• EvenDegreeVertexCountSt000309The vertices with an even number of edges.
• FeedbackVertexSetNumberSt001331The fewest vertices whose deletion leaves a graph with no cycle.
• 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.
• GirthSt000264The length of the shortest cycle.
• GraphComponentCountSt000287The connected components: the vertices that are the first of their component.
• GraphCycleCountSt001736The cycles, each counted once however it is started or turned.
• GraphDifferentialSt001723The most that the vertices next to a set, outside it, exceed the set in number.
• GraphEdgeConnectivitySt000261The fewest edges whose deletion leaves a disconnected graph: the fewest edges between a set of vertices and the rest.
• GraphEdgeCountSt000081The edges of the graph.
• GraphLeafCountSt000312The vertices with exactly one edge.
• GraphLongestCycleLengthSt000637The length of the longest cycle.
• GraphOpenPackingNumberSt000917The most vertices with pairwise disjoint sets of neighbors.
• GraphPackingNumberSt000916The most vertices, no two within distance 2: the closed neighborhoods of the vertices are disjoint.
• GraphTwoLimitedPackingNumberSt000918The most vertices in a set that has at most two members among the closed neighborhood of any vertex.
• GraphVertexConnectivitySt000262The fewest vertices whose deletion leaves a disconnected graph; the complete graph on n vertices gives n - 1.
• GutmanIndexSt000466The sum over the pairs of vertices of the product of their degrees times their distance.
• HalfAlbertsonIndexSt001350Half the sum over the edges of the difference of the degrees of its ends.
• HalfTotalIrregularitySt001521Half the sum over the unordered pairs of vertices of the difference of their degrees.
• HosoyaIndexSt000468The sets of edges with no two sharing a vertex, the empty set included.
• HyperWienerIndexSt000467The sum over the pairs of vertices of their distance plus its square.
• IndependenceGapSt001323The independence number less the fewest vertices of a maximal independent set.
• IndependenceNumberSt000093The most vertices that are pairwise non-adjacent.
• IndependentDominationNumberSt001322The fewest vertices in a set that is independent and dominating: a maximal independent set.
• IndependentSetCountSt000300The sets of vertices, the empty set included, with no two adjacent.
• InducedDiamondCountSt001310The sets of four vertices that induce a diamond: a cycle on four vertices with one chord, five edges.
• InducedFourCycleCountSt001305The sets of four vertices that induce a cycle: every one has exactly two neighbors among the four.
• InducedMatchingNumberSt001393The most edges whose ends induce a subgraph that is just those edges.
• InducedPathOnFourVerticesCountSt001306The sets of four vertices that induce a path: three edges, and every vertex has one or two neighbors among the four.
• InducedPathOnThreeVerticesCountSt001308The sets of three vertices that induce a path: exactly two of the three edges.
• InducedStarOnFourVerticesCountSt001307The sets of four vertices that induce a star: three edges, and one vertex has all three others as neighbors.
• IsolatedVertexCountSt000315The vertices with no edge.
• KingCountSt001691The vertices whose neighbors all have a smaller degree; an isolated vertex counts.
• LargestInducedForestSizeSt001321The most vertices that induce a graph with no cycle.
• LargestInducedStarSizeSt001674The most vertices that induce a star: one vertex adjacent to all the others, which are not adjacent to each other. A single vertex is a star.
• LongestPathLengthSt001120The edges of the longest path.
• MatchingNumberSt000387The most edges with no two sharing a vertex.
• MaximalIndependentSetCountSt001304The independent sets of vertices that no vertex can be added to.
• MaximumCutSizeSt001869The most edges between the two sides of a division of the vertices.
• MaximumDegreeSt000171The greatest number of edges at one vertex.
• MaximumDegreeMultiplicitySt001366The most vertices that share one degree.
• MaximumDegreeVertexCountSt001368The vertices whose degree is the greatest.
• MaximumDistanceSt000259The greatest distance between two vertices: the diameter.
• MaximumLeafNeighborCountSt001826The most neighbors of degree 1 a vertex has.
• MaximumNeighborhoodMultiplicitySt000723The most vertices that have the same set of neighbors.
• MinimalDominatingSetCountSt001302The dominating sets that stop dominating when any vertex is dropped.
• MinimumDegreeSt000310The fewest edges at one vertex.
• MinimumEccentricitySt000260The least eccentricity of a vertex, the greatest distance from it to another vertex: the radius.
• MinimumVertexCoverCountSt000363The vertex covers with the fewest vertices: the independent sets with the most.
• NonTriangleVertexCountSt001996The vertices on no triangle.
• OddDegreeVertexCountSt000311The vertices with an odd number of edges.
• OreDegreeSt000915The greatest sum of the degrees of the two ends of an edge.
• PerfectMatchingCountSt000274The sets of edges that meet every vertex exactly once.
• SameNeighborhoodPairCountSt001347The pairs of vertices that have the same set of neighbors.
• SchultzIndexSt000464The sum over the pairs of vertices of the sum of their degrees times their distance.
• SecondZagrebIndexSt000361The sum over the edges of the product of the degrees of its ends.
• SmallestMissingDegreeSt001367The least number, 0 or more, that is not the degree of any vertex.
• SpanningSubgraphCountSt000343The subgraphs on all the vertices: 2 to the power of the number of edges.
• SpanningTreeCountSt000096The spanning trees: by the matrix-tree theorem, the determinant of the Laplacian with its last row and column deleted.
• SzegedIndexSt000263The sum over the edges of the vertices nearer one end times those nearer the other.
• 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.
• TuranEdgeExcessSt001798The edges less those of the complement of the Turán graph with as many parts as the independence number: disjoint cliques as equal in size as possible.
• UpperDominationNumberSt001337The most vertices in a minimal dominating set.
• 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

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