Identifier
-
Mp00100:
Dyck paths
—touch composition⟶
Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤ
Values
[1,0,1,0] => [1,1] => ([(0,1)],2) => -1
[1,0,1,0,1,0] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => -1
[1,0,1,1,0,0] => [1,2] => ([(1,2)],3) => 0
[1,1,0,0,1,0] => [2,1] => ([(0,2),(1,2)],3) => 0
[1,0,1,0,1,0,1,0] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => -1
[1,0,1,0,1,1,0,0] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
[1,0,1,1,0,1,0,0] => [1,3] => ([(2,3)],4) => 0
[1,0,1,1,1,0,0,0] => [1,3] => ([(2,3)],4) => 0
[1,1,0,0,1,0,1,0] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
[1,1,0,0,1,1,0,0] => [2,2] => ([(1,3),(2,3)],4) => 0
[1,1,0,1,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,1,1,0,0,0,1,0] => [3,1] => ([(0,3),(1,3),(2,3)],4) => 0
[1,0,1,0,1,0,1,0,1,0] => [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
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,0,1,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,0,1,0,1,1,1,0,0,0] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[1,0,1,1,0,1,0,1,0,0] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,0,1,1,0,0,0] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,1,0,0,1,0,0] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,1,0,1,0,0,0] => [1,4] => ([(3,4)],5) => 0
[1,0,1,1,1,1,0,0,0,0] => [1,4] => ([(3,4)],5) => 0
[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,0,0,1,1,0,1,0,0] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,0,1,1,1,0,0,0] => [2,3] => ([(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,0,1,0,0,1,1,0,0] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,1,0,0,0,1,1,0,0] => [3,2] => ([(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,0,1,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [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
[1,0,1,0,1,0,1,0,1,1,0,0] => [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
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,5] => ([(4,5)],6) => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [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
[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4] => ([(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [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
[1,1,0,1,0,0,1,0,1,1,0,0] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,1,0,1,0,0] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,0,1,0,1,0] => [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
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3] => ([(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [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
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [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
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [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
>>> Load all 294 entries. <<<
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.
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.
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
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!