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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000264
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [1,3,5,6,2,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [1,3,6,2,5,4] => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 3
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,5},{2,3,6}}
=> [4,3,6,5,1,2] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,5},{2,3},{6}}
=> [4,3,2,5,1,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,6},{2,3,5}}
=> [4,3,5,6,2,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,4,6},{2,3},{5}}
=> [4,3,2,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,5,6},{2,3,4}}
=> [5,3,4,2,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,5,6},{2,3},{4}}
=> [5,3,2,4,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,6},{2,3,5},{4}}
=> [6,3,5,4,2,1] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,5},{2,6},{3}}
=> [4,6,3,5,1,2] => [1,4,5,2,6,3] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,4,5},{2},{3,6}}
=> [4,2,6,5,1,3] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,5},{2},{3},{6}}
=> [4,2,3,5,1,6] => [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,4,6},{2,5},{3}}
=> [4,5,3,6,2,1] => [1,4,6,2,5,3] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
{{1,4,6},{2},{3,5}}
=> [4,2,5,6,3,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,4,6},{2},{3},{5}}
=> [4,2,3,6,5,1] => [1,4,6,2,3,5] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,5,6},{2,4},{3}}
=> [5,4,3,2,6,1] => [1,5,6,2,4,3] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3
{{1,5},{2,4,6},{3}}
=> [5,4,3,6,1,2] => [1,5,2,4,6,3] => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 3
{{1,5},{2,4},{3,6}}
=> [5,4,6,2,1,3] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,5},{2,4},{3},{6}}
=> [5,4,3,2,1,6] => [1,5,2,4,3,6] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,6},{2,4,5},{3}}
=> [6,4,3,5,2,1] => [1,6,2,4,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,6},{2,4},{3,5}}
=> [6,4,5,2,3,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,6},{2,4},{3},{5}}
=> [6,4,3,2,5,1] => [1,6,2,4,3,5] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,5,6},{2},{3,4}}
=> [5,2,4,3,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,5},{2,6},{3,4}}
=> [5,6,4,3,1,2] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1},{2,5,6},{3,4}}
=> [1,5,4,3,6,2] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,5},{2,6},{3},{4}}
=> [5,6,3,4,1,2] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4
{{1,6},{2,5},{3},{4}}
=> [6,5,3,4,2,1] => [1,6,2,5,3,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1},{2,5,6},{3},{4}}
=> [1,5,3,4,6,2] => [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4
{{1,6},{2},{3,5},{4}}
=> [6,2,5,4,3,1] => [1,6,2,3,5,4] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1},{2,6},{3,5},{4}}
=> [1,6,5,4,3,2] => [1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => [1,2,3,7,4,6,5] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> 3
{{1,2,4,6,7},{3,5}}
=> [2,4,5,6,3,7,1] => [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4
{{1,2,4,6,7},{3},{5}}
=> [2,4,3,6,5,7,1] => [1,2,4,6,7,3,5] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4
{{1,2,4,7},{3,6},{5}}
=> [2,4,6,7,5,3,1] => [1,2,4,7,3,6,5] => ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 3
{{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => [1,2,5,6,7,3,4] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4
{{1,2,5,6},{3,4,7}}
=> [2,5,4,7,6,1,3] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4
{{1,2,5,6},{3,4},{7}}
=> [2,5,4,3,6,1,7] => [1,2,5,6,3,4,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000455
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%
Values
{{1,4,5},{2,3}}
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0 = 4 - 4
{{1,4,5},{2},{3}}
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0 = 4 - 4
{{1,5},{2,4},{3}}
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 4
{{1,2,5,6},{3,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,2,5,6},{3},{4}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,2,6},{3,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,3,5,6},{2,4}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,3,5,6},{2},{4}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,3,6},{2,5},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,4,5,6},{2,3}}
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,5},{2,3,6}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,5},{2,3},{6}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,4,6},{2,3,5}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,6},{2,3},{5}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,5,6},{2,3,4}}
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,5,6},{2,3},{4}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,6},{2,3,5},{4}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,4,5,6},{2},{3}}
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,5},{2,6},{3}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,4,5},{2},{3,6}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,5},{2},{3},{6}}
=> [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 = 4 - 4
{{1,4,6},{2,5},{3}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,4,6},{2},{3,5}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,4,6},{2},{3},{5}}
=> [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 = 4 - 4
{{1,5,6},{2,4},{3}}
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,5},{2,4,6},{3}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,5},{2,4},{3,6}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,6},{2,4,5},{3}}
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,6},{2,4},{3,5}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,6},{2,4},{3},{5}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,5,6},{2},{3,4}}
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0 = 4 - 4
{{1,5},{2,6},{3,4}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,6},{2,5},{3,4}}
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1},{2,5,6},{3,4}}
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,5,6},{2},{3},{4}}
=> [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 = 4 - 4
{{1,5},{2,6},{3},{4}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,6},{2,5},{3},{4}}
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1},{2,5,6},{3},{4}}
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 4
{{1,6},{2},{3,5},{4}}
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1},{2,6},{3,5},{4}}
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 - 4
{{1,2,3,6,7},{4,5}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,3,6,7},{4},{5}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,3,7},{4,6},{5}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,4,6,7},{3,5}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,4,6,7},{3},{5}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,4,7},{3,6},{5}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,5,6,7},{3,4}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,6},{3,4,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,6},{3,4},{7}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,5,7},{3,4,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,7},{3,4},{6}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,6,7},{3,4,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,6,7},{3,4},{5}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,7},{3,4,6},{5}}
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,5,6,7},{3},{4}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,6},{3,7},{4}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,5,6},{3},{4,7}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,2,5,7},{3,6},{4}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,5,7},{3},{4,6}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,2,6,7},{3,5},{4}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,6},{3,5,7},{4}}
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,6},{3,5},{4,7}}
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,6},{3,5},{4},{7}}
=> [3,2,1,1] => ([(0,5),(0,6),(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)
=> ? = 3 - 4
{{1,2,7},{3,5,6},{4}}
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,7},{3,5},{4,6}}
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2,7},{3,5},{4},{6}}
=> [3,2,1,1] => ([(0,5),(0,6),(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)
=> ? = 3 - 4
{{1,2,6,7},{3},{4,5}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,2,6},{3,7},{4,5}}
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,7},{3,6},{4,5}}
=> [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,2},{3,6,7},{4,5}}
=> [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,2,6},{3,7},{4},{5}}
=> [3,2,1,1] => ([(0,5),(0,6),(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)
=> ? = 4 - 4
{{1,2,7},{3,6},{4},{5}}
=> [3,2,1,1] => ([(0,5),(0,6),(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)
=> ? = 3 - 4
{{1,2},{3,6,7},{4},{5}}
=> [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,2,7},{3},{4,6},{5}}
=> [3,1,2,1] => ([(0,6),(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)
=> ? = 3 - 4
{{1,2},{3,7},{4,6},{5}}
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,3,4,6,7},{2,5}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,4,6,7},{2},{5}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,4,7},{2,6},{5}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3 - 4
{{1,3,5,6,7},{2,4}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,6},{2,4,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,6},{2,4},{7}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,3,5,7},{2,4,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,7},{2,4},{6}}
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4 - 4
{{1,3,6,7},{2,4,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,6,7},{2},{4}}
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,6},{2},{4,7}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,6},{2},{4},{7}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,3,5,7},{2},{4,6}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,5,7},{2},{4},{6}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,3,6,7},{2},{4,5}}
=> [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(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 = 4 - 4
{{1,4,5,6,7},{2,3}}
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,4,5,6},{2,3,7}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,4,5,7},{2,3,6}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,4,5},{2,3,6,7}}
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
{{1,4,6,7},{2,3,5}}
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 4 - 4
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.
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