Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001302
(load all 6 compositions to match this statistic)
Mapping
St001302: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 1
([],2)
 => 1
([(0,1)],2)
 => 2
([],3)
 => 1
([(1,2)],3)
 => 2
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 1
([(2,3)],4)
 => 2
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2)],4)
 => 4
([(0,3),(1,2),(2,3)],4)
 => 4
([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 6
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([],5)
 => 1
([(3,4)],5)
 => 2
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
([(1,4),(2,3)],5)
 => 4
([(1,4),(2,3),(3,4)],5)
 => 4
([(0,1),(2,4),(3,4)],5)
 => 4
([(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,3),(1,4),(2,3),(2,4)],5)
 => 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 8
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 7
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 6
([(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)
 => 4
([(0,3),(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)
 => 9
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 7
Description
The number of minimally dominating sets of vertices of a graph. A subset of vertices is '''dominating''' if every vertex is either in this subset or adjacent to an element therein [1]. If a set of vertices is dominating, then so is every superset of this set. This statistic counts the minimally dominating sets.
Matching statistic: St000454
Mapping
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ 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: 27%
Values
([],1)
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
([],2)
 => [2] => [2] => ([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => [1,1] => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => [3] => [3] => ([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [2,1] => [1,2] => ([(1,2)],3)
 => 1 = 2 - 1
([(0,1),(0,2),(1,2)],3)
 => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => [4] => [4] => ([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,1] => [1,3] => ([(2,3)],4)
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 1
([(0,3),(1,2),(2,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 6 - 1
([(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)
 => ? = 3 - 1
([(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)
 => 3 = 4 - 1
([],5)
 => [5] => [5] => ([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4,1] => [1,4] => ([(3,4)],5)
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(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)
 => ? = 3 - 1
([(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)
 => ? = 3 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 6 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 5 - 1
([(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)
 => ? = 3 - 1
([(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 - 1
([(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)
 => ? = 3 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 8 - 1
([(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)
 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 6 - 1
([(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)
 => ? = 6 - 1
([(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 - 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)
 => ? = 5 - 1
([(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)
 => ? = 7 - 1
([(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 - 1
([(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)
 => ? = 6 - 1
([(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 - 1
([(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 - 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)
 => ? = 4 - 1
([(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)
 => ? = 9 - 1
([(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)
 => ? = 7 - 1
([(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 - 1
([(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)
 => 4 = 5 - 1
([],6)
 => [6] => [6] => ([],6)
 => 0 = 1 - 1
([(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
([(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
([(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1] => [1,5] => ([(4,5)],6)
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(1,2),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(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)
 => ? = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(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)
 => ? = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(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)
 => ? = 3 - 1
([(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)
 => ? = 3 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 6 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 5 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => ? = 4 - 1
([(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)
 => ? = 3 - 1
([(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
([(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)
 => 5 = 6 - 1
([],7)
 => [7] => [7] => ([],7)
 => 0 = 1 - 1
([(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
([(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
([(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [6,1] => [1,6] => ([(5,6)],7)
 => 1 = 2 - 1
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.