Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000772
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2] => [1,1] => ([(0,1)],2)
 => 1
{{1,2,3}}
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
{{1},{2,3}}
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 1
{{1,2,3,4}}
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,3},{2,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1,4},{2,3}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
{{1,2,3,4,5}}
 => [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)
 => 4
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,3,4},{2,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2,3,4}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2,3,4,5}}
 => [1,4] => [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},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1},{2,4},{3,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1},{2,5},{3,4}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1,2,3,4,5,6}}
 => [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)
 => 5
{{1,2,3,4},{5,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3,5},{4,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3,6},{4,5}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,3},{4,5,6}}
 => [3,3] => [1,1,2,1,1] => ([(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
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2,4,5},{3,6}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,4,6},{3,5}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,4},{3,5,6}}
 => [3,3] => [1,1,2,1,1] => ([(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
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2,5,6},{3,4}}
 => [4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,5},{3,4,6}}
 => [3,3] => [1,1,2,1,1] => ([(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
{{1,2,6},{3,4,5}}
 => [3,3] => [1,1,2,1,1] => ([(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
{{1,2},{3,4,5,6}}
 => [2,4] => [1,2,1,1,1] => ([(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
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,5},{3},{4,6}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
{{1,2,6},{3},{4,5}}
 => [3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000993
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
{{1}}
 => [1] => [[1],[]]
 => []
 => ? = 1
{{1,2}}
 => [2] => [[2],[]]
 => []
 => ? = 1
{{1,2,3}}
 => [3] => [[3],[]]
 => []
 => ? = 2
{{1},{2,3}}
 => [1,2] => [[2,1],[]]
 => []
 => ? = 1
{{1,2,3,4}}
 => [4] => [[4],[]]
 => []
 => ? = 3
{{1,2},{3,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 1
{{1,3},{2,4}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 1
{{1,4},{2,3}}
 => [2,2] => [[3,2],[1]]
 => [1]
 => ? = 1
{{1},{2,3,4}}
 => [1,3] => [[3,1],[]]
 => []
 => ? = 1
{{1},{2},{3,4}}
 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? = 2
{{1,2,3,4,5}}
 => [5] => [[5],[]]
 => []
 => ? = 4
{{1,2,3},{4,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,2,4},{3,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,2,5},{3,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,2},{3,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
{{1,3,4},{2,5}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,3,5},{2,4}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,3},{2,4,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
{{1,4,5},{2,3}}
 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
{{1,4},{2,3,5}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1
{{1,5},{2,3,4}}
 => [2,3] => [[4,2],[1]]
 => [1]
 => ? = 1
{{1},{2,3,4,5}}
 => [1,4] => [[4,1],[]]
 => []
 => ? = 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
{{1},{2,4},{3,5}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 2
{{1},{2,5},{3,4}}
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? = 2
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? = 3
{{1,2,3,4,5,6}}
 => [6] => [[6],[]]
 => []
 => ? = 5
{{1,2,3,4},{5,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,5},{4,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3,6},{4,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,3},{4,5,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,2,4,5},{3,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4,6},{3,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,4},{3,5,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,2,5,6},{3,4}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,2,5},{3,4,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,2,6},{3,4,5}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,2},{3,4,5,6}}
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,2,5},{3},{4,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,2,6},{3},{4,5}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,2},{3},{4,5,6}}
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2
{{1,2},{3},{4},{5,6}}
 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
{{1,3,4,5},{2,6}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,3,4,6},{2,5}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,3,4},{2,5,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,3,4},{2},{5,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,3,5,6},{2,4}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,3,5},{2,4,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,3,6},{2,4,5}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,3},{2,4,5,6}}
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,3,5},{2},{4,6}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,3,6},{2},{4,5}}
 => [3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 2
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,3},{2},{4,5,6}}
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 2
{{1,3},{2},{4},{5,6}}
 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 3
{{1,4,5,6},{2,3}}
 => [4,2] => [[5,4],[3]]
 => [3]
 => 1
{{1,4,5},{2,3,6}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,4,6},{2,3,5}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,4},{2,3,5,6}}
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1,5,6},{2,3,4}}
 => [3,3] => [[5,3],[2]]
 => [2]
 => 1
{{1,5},{2,3,4,6}}
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
{{1,6},{2,3,4,5}}
 => [2,4] => [[5,2],[1]]
 => [1]
 => ? = 1
{{1},{2,3,4,5,6}}
 => [1,5] => [[5,1],[]]
 => []
 => ? = 1
{{1},{2,3,4},{5,6}}
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
{{1},{2,3},{4,5,6}}
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,4},{3,5,6}}
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,5},{3,4,6}}
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,6},{3,4,5}}
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2},{3,4,5,6}}
 => [1,1,4] => [[4,1,1],[]]
 => []
 => ? = 2
{{1},{2},{3,4},{5,6}}
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? = 1
{{1},{2},{3,5},{4,6}}
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? = 1
{{1},{2},{3,6},{4,5}}
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => ? = 1
{{1},{2},{3},{4,5,6}}
 => [1,1,1,3] => [[3,1,1,1],[]]
 => []
 => ? = 3
{{1},{2},{3},{4},{5,6}}
 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
 => []
 => ? = 4
{{1,2,3,4,5,6,7}}
 => [7] => [[7],[]]
 => []
 => ? = 6
{{1,2},{3,4,5,6,7}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1,3},{2,4,5,6,7}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1,4},{2,3,5,6,7}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1,5},{2,3,4,6,7}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1,6},{2,3,4,5,7}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1,7},{2,3,4,5,6}}
 => [2,5] => [[6,2],[1]]
 => [1]
 => ? = 1
{{1},{2,3,4,5,6,7}}
 => [1,6] => [[6,1],[]]
 => []
 => ? = 1
{{1},{2,3},{4,5,6,7}}
 => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,4},{3,5,6,7}}
 => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,5},{3,4,6,7}}
 => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1
{{1},{2,6},{3,4,5,7}}
 => [1,2,4] => [[5,2,1],[1]]
 => [1]
 => ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000455
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 21%●distinct values known / distinct values provided: 17%
Values
{{1}}
 => [1] => [1] => ([],1)
 => ? = 1 - 1
{{1,2}}
 => [2] => [2] => ([],2)
 => ? = 1 - 1
{{1,2,3}}
 => [3] => [3] => ([],3)
 => ? = 2 - 1
{{1},{2,3}}
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 1 - 1
{{1,2,3,4}}
 => [4] => [4] => ([],4)
 => ? = 3 - 1
{{1,2},{3,4}}
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,3},{2,4}}
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1,4},{2,3}}
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2,3,4}}
 => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0 = 1 - 1
{{1},{2},{3,4}}
 => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 - 1
{{1,2,3,4,5}}
 => [5] => [5] => ([],5)
 => ? = 4 - 1
{{1,2,3},{4,5}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,4},{3,5}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2,5},{3,4}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3,4,5}}
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
{{1,3,4},{2,5}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3,5},{2,4}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2,4,5}}
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,3},{2},{4,5}}
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
{{1,4,5},{2,3}}
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,4},{2,3,5}}
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1,5},{2,3,4}}
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1},{2,3,4,5}}
 => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
{{1},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,4},{2},{3,5}}
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
{{1},{2,4},{3,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1,5},{2},{3,4}}
 => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
{{1},{2,5},{3,4}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 - 1
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 - 1
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
{{1,2,3,4,5,6}}
 => [6] => [6] => ([],6)
 => ? = 5 - 1
{{1,2,3,4},{5,6}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,3,5},{4,6}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,3,6},{4,5}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,3},{4,5,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,3},{4},{5,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2,4,5},{3,6}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,4,6},{3,5}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,4},{3,5,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,4},{3},{5,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2,5,6},{3,4}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,5},{3,4,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2,6},{3,4,5}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2},{3,4,5,6}}
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,2},{3,4},{5,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,5},{3},{4,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2},{3,5},{4,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,6},{3},{4,5}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2},{3,6},{4,5}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2},{3},{4,5,6}}
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,2},{3},{4},{5,6}}
 => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
{{1,3,4,5},{2,6}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,4,6},{2,5}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,4},{2,5,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,4},{2},{5,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,3,5,6},{2,4}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,5},{2,4,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3,6},{2,4,5}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3},{2,4,5,6}}
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,3},{2,4},{5,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,3,5},{2},{4,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,3},{2,5},{4,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,3,6},{2},{4,5}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,3},{2,6},{4,5}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,3},{2},{4,5,6}}
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,3},{2},{4},{5,6}}
 => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
{{1,4,5,6},{2,3}}
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4,5},{2,3,6}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4,6},{2,3,5}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4},{2,3,5,6}}
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,4},{2,3},{5,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,5,6},{2,3,4}}
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,5},{2,3,4,6}}
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1,6},{2,3,4,5}}
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3,4,5,6}}
 => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
{{1},{2,3,4},{5,6}}
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,5},{2,3},{4,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1},{2,3,5},{4,6}}
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,6},{2,3},{4,5}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1},{2,3,6},{4,5}}
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1},{2,3},{4,5,6}}
 => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1},{2,3},{4},{5,6}}
 => [1,2,1,2] => [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)
 => ? = 2 - 1
{{1,4,5},{2},{3,6}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,4},{2,5},{3,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,4,6},{2},{3,5}}
 => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,4},{2,6},{3,5}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,4},{2},{3,5,6}}
 => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 - 1
{{1,4},{2},{3},{5,6}}
 => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 - 1
{{1,5},{2,4},{3,6}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1},{2,4,5},{3,6}}
 => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,6},{2,4},{3,5}}
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 - 1
{{1,2,3,4,5},{6,7}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,4,6},{5,7}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,4,7},{5,6}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,4},{5,6,7}}
 => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,5,6},{4,7}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,5,7},{4,6}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,5},{4,6,7}}
 => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
{{1,2,3,6,7},{4,5}}
 => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0 = 1 - 1
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.