MaximumNeighborhoodMultiplicity

The most vertices that have the same set of neighbors.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000723via LabeledGraph✓↗

MaximumNeighborhoodMultiplicity(graph)The most vertices that have the same set of neighbors.

Typed signatures

Type	Package	Applies
(labeled_graph) -> number	combinatorics	always

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 St000723. 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(
                                                                                        (adjacency: list<list<integer>>) => Max(Map(
                                                                                                                                     Function(Max(Sum(Map(Function(1 if Max(Sum(Map(Function(Abs(adjacency[u][x] - adjacency[v][x]), x), Range(1, n))), 0) ==
                                                                                                                                                                        0 else 0, u), Range(1, n))), 0), v),
                                                                                                                                     Range(1, n),
                                                                                                                                   )),
                                                                                        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)),
                                                                                      ), _x[2]), _x[1])

Examples

Basic #

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

live ↗

In$\mathrm{MaximumNeighborhoodMultiplicity}(\operatorname{LabeledGraph}((4,\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)))$Out$1$

Properties #

FindStat's value for ([],1)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((1,\bigl\lbrack\bigr\rbrack)), \text{MaximumNeighborhoodMultiplicity})$Out$1$

Known value · FindStat St000723

FindStat's value for ([],2)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((2,\bigl\lbrack\bigr\rbrack)), \text{MaximumNeighborhoodMultiplicity})$Out$2$

Known value · FindStat St000723

FindStat's value for ([],3)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((3,\bigl\lbrack\bigr\rbrack)), \text{MaximumNeighborhoodMultiplicity})$Out$3$

Known value · FindStat St000723

About

Examples	4
Crosswalk	FindStat