Identifier
Values
[2] => [1,1] => ([(0,1)],2) => ([(1,2)],3) => 0
[1,2] => [1,2] => ([(1,2)],3) => ([(2,3)],4) => 0
[2,1] => [2,1] => ([(0,2),(1,2)],3) => ([(1,3),(2,3)],4) => 0
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(1,2),(1,3),(2,3)],4) => 0
[1,1,2] => [1,3] => ([(2,3)],4) => ([(3,4)],5) => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4) => ([(2,4),(3,4)],5) => 0
[1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(2,3),(2,4),(3,4)],5) => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => ([(1,4),(2,4),(3,4)],5) => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[1,1,1,2] => [1,4] => ([(3,4)],5) => ([(4,5)],6) => 0
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5) => ([(3,5),(4,5)],6) => 0
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(3,4),(3,5),(4,5)],6) => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5) => ([(2,5),(3,5),(4,5)],6) => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[5] => [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) => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[1,1,1,1,2] => [1,5] => ([(4,5)],6) => ([(5,6)],7) => 0
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6) => ([(4,6),(5,6)],7) => 0
[1,1,1,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(4,5),(4,6),(5,6)],7) => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6) => ([(3,6),(4,6),(5,6)],7) => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,1,4] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[1,5] => [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) => ([(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] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 0
[4,1,1] => [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) => ([(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
[5,1] => [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) => ([(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
[6] => [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) => ([(1,2),(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
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
vertex addition
Description
Adds a disconnected vertex to a graph.
Map
conjugate
Description
The conjugate of a composition.
The conjugate of a composition $C$ is defined as the complement (Mp00039complement) of the reversal (Mp00038reverse) of $C$.
Equivalently, the ribbon shape corresponding to the conjugate of $C$ is the conjugate of the ribbon shape of $C$.
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.