Identifier
Values
[.,.] => ([],1) => ([],1) => ([(0,1)],2) => -1
[.,[.,.]] => ([(0,1)],2) => ([],2) => ([(0,2),(1,2)],3) => 0
[[.,.],.] => ([(0,1)],2) => ([],2) => ([(0,2),(1,2)],3) => 0
[.,[.,[.,.]]] => ([(0,2),(2,1)],3) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[.,[[.,.],.]] => ([(0,2),(2,1)],3) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[[.,[.,.]],.] => ([(0,2),(2,1)],3) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[[[.,.],.],.] => ([(0,2),(2,1)],3) => ([],3) => ([(0,3),(1,3),(2,3)],4) => 0
[.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => ([],4) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0
[.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([],5) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[.,[[[[[.,.],.],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[.,[[.,[.,.]],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[.,[[.,.],.]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[.,[.,.]],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[.,[[[[.,.],.],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[.,[.,[.,.]]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[.,[[[.,.],.],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,[.,[.,.]]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[.,[[.,.],.]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[[.,[.,.]],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[[[[[[.,.],.],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([],6) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
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
cone
Description
The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
Map
incomparability graph
Description
The incomparability graph of a poset.