TuranEdgeExcess
The 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.
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St001798via LabeledGraph✓TuranEdgeExcess(graph)The 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.Typed signatures
| Type | Package | Applies |
|---|---|---|
(labeled_graph) -> number | combinatorics | always |
Details
- Defined over
LabeledGraphas an expression in_x, evaluated by compute-engine — the definition IS the implementation. - Takes a
LabeledGraphand 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 St001798, nonnegative by Turán's theorem. The independence number is the number of cliques; n mod it of them have one more vertex than the rest. 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(
(inside: list<integer>) => Apply(
(totals: list<integer>) => Apply(
(parts: integer) => Apply((extra: integer) => Apply((size: integer) => Length(edges) -
(extra *
(size + 1) *
size +
(parts - extra) *
size *
(size - 1)) /
2, (n - extra) /
parts), n % parts),
Max(Map(Function((1 if inside[m + 1] ==
0 else 0) *
totals[m + 1], m), Range(0, 2 ^ n -
1))),
),
Map(
Function(Max(Sum(Map(Function(1 if m %
2 ^ v >=
2 ^
(v - 1) else 0, v), Range(1, n))), 0), m),
Range(0, 2 ^ n -
1),
),
),
Map(
Function(Max(Sum(Map(Function((1 if m %
2 ^ e[1] >=
2 ^
(e[1] - 1) else 0) *
(1 if m %
2 ^ e[2] >=
2 ^
(e[2] - 1) else 0), e), edges)), 0), m),
Range(0, 2 ^ n -
1),
),
), _x[2]), _x[1])Examples
Properties #
About
| Examples | 4 |
|---|---|
| Crosswalk | FindStat |