- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001373

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001373: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [.,.]
 => [1] => ([],1)
 => 1
[1,2] => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[2,1] => [[.,.],.]
 => [1,2] => ([],2)
 => 2
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 3
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => 2
[2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => 2
[3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => 2
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => 3
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 4
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 4
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 3
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 3
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => 3
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 4
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => 3
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => 3
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => 3
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => 4
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3

Description

The logarithm of the number of winning configurations of the lights out game on a graph. In the single player lamps out game, every vertex has two states, on or off. The player can toggle the state of a vertex, in which case all the neighbours of the vertex change state, too. The goal is to reach the configuration with all vertices off.

Matching statistic: St000454

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest eigenvalue of a graph if it is integral. If a graph is

$d$ -regular, then its largest eigenvalue equals

$d$ . One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.