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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001060
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,2,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Mp00315: Integer compositions —inverse Foata bijection⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,2] => [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)
=> 4
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,2,1] => [2,1,1,1,1] => ([(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)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,2,1,1] => [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)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,2,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,1,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,2,1] => [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)
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,1,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => [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)
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,2] => [1,1,1,1,1,2] => ([(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)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,2,1] => [2,1,1,1,1,1] => ([(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)
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,3] => [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)
=> 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,3] => [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)
=> 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,2,1,1] => [1,2,1,1,1,1] => ([(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)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,2,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,3,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,3,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(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)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,2,1,2] => [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)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,2,2,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(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)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,3,1,1] => [1,3,1,1,1] => ([(0,4),(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),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,3,2] => [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)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,4,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,4,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,3,1,1] => [1,3,1,1,1] => ([(0,4),(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),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,3,2] => [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)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,4,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,4,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,4,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2
[1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [3,1,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [3,1,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [3,1,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
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.
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