Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000259
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Mapping
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => 0
[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 3
[1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 5
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 4
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 4
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 4
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 5
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 4
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 4
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000455
Mapping
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 14%
Values
[1,0]
 => [[1],[]]
 => ([],1)
 => ([],1)
 => ? = 0 - 3
[1,0,1,0]
 => [[1,1],[]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 1 - 3
[1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 3
[1,1,0,1,0,0]
 => [[3],[]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 3
[1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 3
[1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 3 - 3
[1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,0]
 => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 2 - 3
[1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 2 - 3
[1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 2 - 3
[1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 3 - 3
[1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
[1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 3
[1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 2 - 3
[1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ([(0,2),(0,3),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
 => ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
[1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 3 - 3
[1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
 => ([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 3 - 3
[1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 - 3
[1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2 - 3
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1,1],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 2 - 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 3 - 3
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1,1],[2]]
 => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1],[]]
 => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 3 - 3
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
 => ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 - 3
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1],[2,2]]
 => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [[4,4,1],[3]]
 => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
 => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1],[]]
 => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,3),(0,6),(1,4),(1,6),(4,2),(4,5),(6,5)],7)
 => ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 5 - 3
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3,1],[1,1]]
 => ([(0,4),(0,7),(1,2),(1,3),(2,5),(3,5),(3,7),(5,6),(7,6)],8)
 => ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3 - 3
[1,0,1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,1],[1]]
 => ([(0,7),(1,4),(1,7),(3,2),(3,6),(4,3),(4,5),(5,6),(7,5)],8)
 => ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 4 - 3
[1,0,1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2,1],[]]
 => ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 4 - 3
[1,0,1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,1],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
 => ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 3
[1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
 => ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
 => ? = 3 - 3
[1,1,0,1,0,0,1,0,1,0,1,0]
 => [[3,3,3,3],[2,2,2]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 3 - 3
[1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 4 - 3
[1,1,0,1,0,1,0,0,1,0,1,0]
 => [[4,4,4],[3,3]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,0,1,0]
 => [[5,5],[4]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 2 - 3
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [[5,4],[2]]
 => ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
 => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [[4,4,4],[2,2]]
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
 => ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3 - 3
[1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,4),(1,5),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
 => ([(0,4),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 5 - 3
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [[5,3],[1]]
 => ([(0,6),(1,4),(1,6),(3,2),(4,3),(4,5),(6,5)],7)
 => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 4 - 3
[1,1,0,1,1,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1,1]]
 => ([(0,7),(1,6),(2,3),(2,7),(3,5),(3,6),(5,4),(6,4),(7,5)],8)
 => ([(1,5),(1,7),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(6,7)],8)
 => ? = 4 - 3
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ([(0,7),(1,3),(1,4),(3,5),(3,7),(4,2),(4,5),(5,6),(7,6)],8)
 => ([(0,7),(1,6),(2,3),(2,4),(2,6),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3 - 3
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
 => ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
 => ? = 3 - 3
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,6),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 3
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ([(0,8),(1,4),(1,8),(2,3),(2,7),(3,5),(4,2),(4,6),(6,7),(7,5),(8,6)],9)
 => ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 3
[1,1,1,0,1,0,0,0,1,0,1,0]
 => [[2,2,2,2,2],[1,1]]
 => ([(0,3),(1,4),(1,7),(2,6),(3,7),(4,2),(4,5),(5,6),(7,5)],8)
 => ([(1,7),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 3 - 3
[1,1,1,0,1,0,0,1,0,0,1,0]
 => [[3,3,2,2],[2]]
 => ([(0,7),(1,3),(1,4),(2,6),(3,5),(3,7),(4,2),(4,5),(5,6)],8)
 => ([(0,7),(1,6),(1,7),(2,3),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3 - 3
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
 => ([(1,6),(1,7),(2,4),(2,5),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 3 - 3
[1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([(0,8),(1,4),(1,8),(2,3),(2,7),(3,5),(4,2),(4,6),(6,7),(7,5),(8,6)],9)
 => ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 3
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [[3,2,2,2],[]]
 => ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
 => ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
 => ? = 3 - 3
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [[2,2,2,2,2],[]]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ([(2,9),(3,8),(4,7),(4,8),(5,6),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 2 - 3
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [[4,3,2],[]]
 => ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
 => ([(1,5),(1,8),(2,3),(2,7),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4 - 3
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [[3,3,3,2],[1]]
 => ([(0,3),(0,9),(1,4),(1,9),(2,5),(3,2),(3,8),(4,7),(7,6),(8,5),(8,6),(9,7),(9,8)],10)
 => ([(0,5),(0,9),(1,4),(1,8),(2,6),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,9),(5,7),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 4 - 3
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [[4,4,2],[]]
 => ([(0,4),(0,5),(1,8),(2,7),(3,2),(3,9),(4,3),(4,6),(5,1),(5,6),(6,8),(6,9),(9,7)],10)
 => ([(1,7),(1,9),(2,5),(2,9),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,8),(6,7),(6,9),(7,8),(8,9)],10)
 => ? = 3 - 3
[1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([(0,2),(0,3),(1,7),(2,8),(3,4),(3,8),(4,6),(4,7),(6,5),(7,5),(8,6)],9)
 => ([(1,8),(2,6),(2,8),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(7,8)],9)
 => ? = 3 - 3
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [[5,4],[]]
 => ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
 => ([(1,8),(2,7),(3,6),(3,7),(4,5),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 3 - 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0 = 3 - 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [[4,1,1,1],[]]
 => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
 => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 0 = 3 - 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [[5,1,1],[]]
 => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0 = 3 - 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6,1],[]]
 => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
[1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [[3,3,3,3,3],[2,2,2,2]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [[4,4,4,4],[3,3,3]]
 => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
 => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [[5,5,5],[4,4]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 0 = 3 - 3
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [[6,6],[5]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 3 - 3
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.