- FindStat

Identifier

Mp00148: Finite Cartan types —to root poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00117: Graphs —Ore closure⟶ Graphs
St001646: Graphs ⟶ ℤ

Values

['A',1] => ([],1) => ([],1) => ([],1) => 0
['A',2] => ([(0,2),(1,2)],3) => ([(1,2)],3) => ([(1,2)],3) => 0
['B',2] => ([(0,3),(1,3),(3,2)],4) => ([(2,3)],4) => ([(2,3)],4) => 1
['G',2] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => ([(4,5)],6) => ([(4,5)],6) => 2
['A',3] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0

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$F_{1} = 1$

$F_{2} = 1 + q + q^{2}$

Description

The number of edges that can be added without increasing the maximal degree of a graph.
This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on

$n$ vertices, which has maximal degree

$n-1$ and therefore has statistic

$\binom{n-1}{2}$ .

Map

Ore closure

Description

The Ore closure of a graph.
The Ore closure of a connected graph

$G$ has the same vertices as

$G$ , and the smallest set of edges containing the edges of

$G$ such that for any two vertices

$u$ and

$v$ whose sum of degrees is at least the number of vertices, then

$(u,v)$ is also an edge.
For disconnected graphs, we compute the closure separately for each component.

Map

incomparability graph

Description

The incomparability graph of a poset.

Map

to root poset

Description

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where

$\alpha \prec \beta$ if

$\beta - \alpha$ is a simple root.