Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001486
(load all 8 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => 1
([(0,1)],2)
 => [1,1] => [1,1] => 2
([(1,2)],3)
 => [2,1] => [1,2] => 3
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => 2
([(2,3)],4)
 => [3,1] => [1,3] => 3
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => 2
([(3,4)],5)
 => [4,1] => [1,4] => 3
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,2,1,1] => 4
([(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] => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => 5
([(0,2),(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] => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => 2
([(4,5)],6)
 => [5,1] => [1,5] => 3
([(3,5),(4,5)],6)
 => [5,1] => [1,5] => 3
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => 3
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => 4
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St000453
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000453: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => ([],1)
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 2
([(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(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)
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(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)
 => 4
([(0,4),(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)
 => 4
([(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)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(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)
 => 4
([(0,3),(0,4),(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)
 => 4
([(0,1),(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)
 => 5
([(0,3),(1,2),(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)
 => 5
([(0,3),(0,4),(1,2),(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)
 => 5
([(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)
 => 5
([(0,1),(0,4),(1,3),(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)
 => 5
([(0,3),(0,4),(1,2),(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)
 => 5
([(0,4),(1,2),(1,3),(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)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(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)
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [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)
 => 2
([(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
Description
The number of distinct Laplacian eigenvalues of a graph.
Matching statistic: St000777
(load all 2 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1] => ([],1)
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 2
([(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,2),(1,2)],3)
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
([(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(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)
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(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)
 => 4
([(0,4),(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)
 => 4
([(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)
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(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)
 => 4
([(0,3),(0,4),(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)
 => 4
([(0,1),(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)
 => 5
([(0,3),(1,2),(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)
 => 5
([(0,3),(0,4),(1,2),(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)
 => 5
([(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)
 => 5
([(0,1),(0,4),(1,3),(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)
 => 5
([(0,3),(0,4),(1,2),(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)
 => 5
([(0,4),(1,2),(1,3),(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)
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(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)
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [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)
 => 2
([(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 3
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001488
(load all 3 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 71%
Values
([],1)
 => [1] => [1] => [[1],[]]
 => 1
([(0,1)],2)
 => [1,1] => [1,1] => [[1,1],[]]
 => 2
([(1,2)],3)
 => [2,1] => [2,1] => [[2,2],[1]]
 => 3
([(0,2),(1,2)],3)
 => [2,1] => [2,1] => [[2,2],[1]]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => [[1,1,1],[]]
 => 2
([(2,3)],4)
 => [3,1] => [3,1] => [[3,3],[2]]
 => 3
([(1,3),(2,3)],4)
 => [3,1] => [3,1] => [[3,3],[2]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [3,1] => [[3,3],[2]]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => 4
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => [[2,2,1],[1]]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => [[1,1,1,1],[]]
 => 2
([(3,4)],5)
 => [4,1] => [4,1] => [[4,4],[3]]
 => 3
([(2,4),(3,4)],5)
 => [4,1] => [4,1] => [[4,4],[3]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => [[4,4],[3]]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [4,1] => [[4,4],[3]]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => [[3,3,1],[2]]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,2,1] => [[2,2,1,1],[1]]
 => 4
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => 2
([(4,5)],6)
 => [5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
([(3,5),(4,5)],6)
 => [5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [5,1] => [[5,5],[4]]
 => ? = 3
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => [[4,4,1],[3]]
 => ? = 4
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,1,1,1] => [1,1,3,1] => [[3,3,1,1],[2]]
 => ? = 4
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [3,2,1] => [[4,4,3],[3,2]]
 => ? = 5
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St001570
Mapping
Mp00156: Graphs —line graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
([],1)
 => ([],0)
 => ([],0)
 => ([],0)
 => ? = 1 - 5
([(0,1)],2)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 5
([(1,2)],3)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 3 - 5
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 2 - 5
([(2,3)],4)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 3 - 5
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 3 - 5
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 4 - 5
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => ? = 2 - 5
([(3,4)],5)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 3 - 5
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 3 - 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? = 3 - 5
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 4 - 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ? = 5 - 5
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 5 - 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ?
 => ? = 5 - 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 5 - 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ?
 => ? = 4 - 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ?
 => ? = 4 - 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ?
 => ? = 5 - 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
 => ?
 => ? = 5 - 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
 => ?
 => ? = 4 - 5
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
 => ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
 => ?
 => ? = 2 - 5
([(4,5)],6)
 => ([],1)
 => ([],1)
 => ([],1)
 => ? = 3 - 5
([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([],2)
 => ([],1)
 => ? = 3 - 5
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 3 - 5
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],4)
 => ([],1)
 => ? = 3 - 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],5)
 => ([],1)
 => ? = 3 - 5
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => ([],3)
 => ([],1)
 => ? = 4 - 5
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => ([(0,1)],2)
 => ? = 4 - 5
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 4 - 5
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? = 4 - 5
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? = 4 - 5
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ?
 => ? = 4 - 5
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
 => ?
 => ? = 4 - 5
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0 = 5 - 5
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
 => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 0 = 5 - 5
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0 = 5 - 5
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000455
(load all 3 compositions to match this statistic)
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 29%
Values
([],1)
 => [1] => [1] => ([],1)
 => ? = 1 - 3
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => -1 = 2 - 3
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 0 = 3 - 3
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 0 = 3 - 3
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 2 - 3
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 0 = 3 - 3
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 4 - 3
([(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)
 => ? = 4 - 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => -1 = 2 - 3
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 0 = 3 - 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 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)
 => ? = 4 - 3
([(0,4),(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)
 => ? = 4 - 3
([(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)
 => ? = 4 - 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4 - 3
([(0,4),(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)
 => ? = 4 - 3
([(0,3),(0,4),(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)
 => ? = 4 - 3
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(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)
 => ? = 5 - 3
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(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)
 => ? = 4 - 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)
 => ? = 4 - 3
([(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)
 => ? = 4 - 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5 - 3
([(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)
 => ? = 5 - 3
([(0,2),(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)
 => ? = 4 - 3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [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)
 => -1 = 2 - 3
([(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 0 = 3 - 3
([(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 0 = 3 - 3
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 0 = 3 - 3
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 0 = 3 - 3
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 0 = 3 - 3
([(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 - 3
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 5 - 3
([(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)
 => [1,1,1,1,1,1] => [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)
 => -1 = 2 - 3
([(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 0 = 3 - 3
([(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)
 => [1,1,1,1,1,1,1] => [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)
 => -1 = 2 - 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.