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CelebrityCount ​

The vertices of whom more than half the neighbors have a smaller degree.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001689via LabeledGraph✓
CelebrityCount(graph)The vertices of whom more than half the neighbors have a smaller degree.
Typed signatures
TypePackageApplies
(labeled_graph) -> numbercombinatoricsalways

Domain: Graph statistics

Details
  • Defined over LabeledGraph as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
  • Takes a LabeledGraph and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
  • FindStat's St001689. The graph with no vertices gives 0.

Definition

Epsil, as the implementation runs it.

on LabeledGraph, over _x

0 if _x[1] == 0 else Apply((n: integer) => Apply((edges: list<list<integer>>) => Apply(
                                                                                        (degrees: list<integer>) => Apply(
                                                                                                                           (adjacency: list<list<integer>>) => Max(Sum(Map(Function(1 if 2 *
                                                                                                                                                                                         Max(Sum(Map(Function(adjacency[v][w] *
                                                                                                                                                                                                              (1 if degrees[w] < degrees[v] else 0), w), Range(1, n))), 0) >
                                                                                                                                                                                         degrees[v] else 0, v), Range(1, n))), 0),
                                                                                                                           Map(
                                                                                                                                Function(Map(Function(Max(Sum(Map(Function((1 if e[1] == i else 0) *
                                                                                                                                                                           (1 if e[2] == j else 0) +
                                                                                                                                                                           (1 if e[1] == j else 0) *
                                                                                                                                                                           (1 if e[2] == i else 0), e), edges)), 0), j), Range(1, n)), i),
                                                                                                                                Range(1, n),
                                                                                                                              ),
                                                                                                                         ),
                                                                                        Map(
                                                                                             Function(Max(Sum(Map(Function((1 if e[1] == v else 0) +
                                                                                                                           (1 if e[2] == v else 0), e), edges)), 0), v),
                                                                                             Range(1, n),
                                                                                           ),
                                                                                      ), _x[2]), _x[1])

Examples

Basic #

the graph on 4 vertices with the edges 12, 23, 24 and 34

live ↗
InCelebrityCount(LabeledGraph⁡((4,[[1,2],[2,3],[2,4],[3,4]])))\mathrm{CelebrityCount}(\operatorname{LabeledGraph}((4,\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)))Out11
Properties #

FindStat's value for ([],1)

live ↗
InCombinatorialStat(LabeledGraph⁡((1,[])),CelebrityCount)\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((1,\bigl\lbrack\bigr\rbrack)), \text{CelebrityCount})Out00

Known value · FindStat St001689

FindStat's value for ([],2)

live ↗
InCombinatorialStat(LabeledGraph⁡((2,[])),CelebrityCount)\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((2,\bigl\lbrack\bigr\rbrack)), \text{CelebrityCount})Out00

Known value · FindStat St001689

FindStat's value for ([],3)

live ↗
InCombinatorialStat(LabeledGraph⁡((3,[])),CelebrityCount)\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((3,\bigl\lbrack\bigr\rbrack)), \text{CelebrityCount})Out00

Known value · FindStat St001689

About
Examples4
CrosswalkFindStat