Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000506
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
([(1,4),(2,3)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [[4,3,1],[2]]
 => [2]
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [[4,4,1],[3]]
 => [3]
 => 0
([(2,5),(3,4)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => 0
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [[5,2],[1]]
 => [1]
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [[5,3],[2]]
 => [2]
 => 0
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St000455
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 4% ●values known / values provided: 7%●distinct values known / distinct values provided: 4%
Values
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 0
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 0
([(1,4),(2,3)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
([(2,5),(3,4)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [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
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [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
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [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)
 => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [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
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [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
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,4),(2,3)],6)
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
([(0,1),(2,5),(3,4),(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)
 => ? = 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(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)
 => ? = 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [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
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(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)
 => ? = 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [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
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,2] => [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)
 => ? = 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [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
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [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
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [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
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [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
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [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)
 => ? = 0
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,1,1,1] => [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
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 0
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0
([(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] => [1,5] => ([(4,5)],6)
 => 0
([(3,6),(4,6),(5,6)],7)
 => [1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(2,6),(3,6),(4,6),(5,6)],7)
 => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 0
([(3,6),(4,5)],7)
 => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
([(4,5),(4,6),(5,6)],7)
 => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
([(1,2),(3,6),(4,6),(5,6)],7)
 => [1,1,2,3] => [3,1,1,2] => ([(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
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,3] => [3,1,1,2] => ([(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
([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(3,5),(3,6),(4,5),(4,6)],7)
 => [1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,2,2] => [2,1,1,1,2] => ([(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)
 => 0
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
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.
Matching statistic: St000454
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 7%
Values
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0 + 2
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
([(0,3),(1,2)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(1,4),(2,3)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
([(2,5),(3,5),(4,5)],6)
 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(2,5),(3,4)],6)
 => [2,4] => [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)
 => ? = 0 + 2
([(3,4),(3,5),(4,5)],6)
 => [2,4] => [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)
 => ? = 0 + 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,5),(1,5),(2,4),(3,4)],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 + 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 0 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(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 + 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [2,3,1] => [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)
 => ? = 1 + 2
([(0,5),(1,4),(2,3)],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)
 => ? = 0 + 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(1,4),(1,5),(2,3),(2,5),(3,4)],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 + 2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(0,6),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,3),(0,4),(0,6),(1,2),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,2),(1,4),(2,4),(3,5),(3,6),(4,5),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(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)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(4,5)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,6),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => [1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
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.