- FindStat

Identifier

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

[2] => [1,1] => [1,1] => ([(0,1)],2) => -1
[1,2] => [2,1] => [1,2] => ([(1,2)],3) => 0
[2,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3) => 0
[3] => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => -1
[1,1,2] => [3,1] => [1,3] => ([(2,3)],4) => 0
[1,2,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,3] => [2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
[2,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[2,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[4] => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => -1
[1,1,1,2] => [4,1] => [1,4] => ([(3,4)],5) => 0
[1,1,2,1] => [3,2] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,3] => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,2,1,1] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,2,2] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,4] => [2,1,1,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[2,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[2,1,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[2,3] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[5] => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -1
[1,1,1,1,2] => [5,1] => [1,5] => ([(4,5)],6) => 0
[1,1,1,2,1] => [4,2] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,1,3] => [4,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,1,2,1,1] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,2,2] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,4] => [3,1,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,2,1,1,1] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,2,1,2] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,2,3] => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,5] => [2,1,1,1,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,1,1,1,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[2,1,1,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,1,3] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,4] => [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[6] => [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1
[1,1,1,1,1,2] => [6,1] => [1,6] => ([(5,6)],7) => 0
[1,1,1,1,2,1] => [5,2] => [2,5] => ([(4,6),(5,6)],7) => 0
[1,1,1,1,3] => [5,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,1,2,1,1] => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,1,2,2] => [4,2,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,1,4] => [4,1,1,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,2,1,1,1] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,2,1,2] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,2,3] => [3,2,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,5] => [3,1,1,1,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,2,1,1,1,1] => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,2,1,1,2] => [2,4,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,2,1,3] => [2,3,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,2,4] => [2,2,1,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,6] => [2,1,1,1,1,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[2,1,1,1,1,1] => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[2,1,1,1,2] => [1,5,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[2,1,1,3] => [1,4,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[2,1,4] => [1,3,1,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[2,5] => [1,2,1,1,1,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[7] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => -1

search for individual values

searching the database for the individual values of this statistic

Description

The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.

Map

to threshold graph

Description

The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.

Map

complement

Description

The complement of a composition.
The complement of a composition

$I$ is defined as follows:
If

$I$ is the empty composition, then the complement is also the empty composition. Otherwise, let

$S$ be the descent set corresponding to

$I=(i_1,\dots,i_k)$ , that is, the subset

$\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$
of

$\{ 1, 2, \ldots, |I|-1 \}$ . Then, the complement of

$I$ is the composition of the same size as

$I$ , whose descent set is

$\{ 1, 2, \ldots, |I|-1 \} \setminus S$ .
The complement of a composition

$I$ coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to

$I$ .

Map

rotate front to back

Description

The front to back rotation of the entries of an integer composition.