- FindStat

Your data matches 79 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000081
(load all 4 compositions to match this statistic)

Mapping

St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => 0
([],2)
 => 0
([(0,1)],2)
 => 1
([],3)
 => 0
([(1,2)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([(0,1),(0,2),(1,2)],3)
 => 3
([],4)
 => 0
([(2,3)],4)
 => 1
([(1,3),(2,3)],4)
 => 2
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 2
([(0,3),(1,2),(2,3)],4)
 => 3
([(1,2),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 6
([],5)
 => 0
([(3,4)],5)
 => 1
([(2,4),(3,4)],5)
 => 2
([(1,4),(2,4),(3,4)],5)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
([(1,4),(2,3)],5)
 => 2
([(1,4),(2,3),(3,4)],5)
 => 3
([(0,1),(2,4),(3,4)],5)
 => 3
([(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)
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 6
([(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)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(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)
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 8

Description

The number of edges of a graph.

Matching statistic: St000228
(load all 13 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 6
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 8

Description

The size of a partition. This statistic is the constant statistic of the level sets.

Matching statistic: St000448
(load all 3 compositions to match this statistic)

Mapping

Mp00203: Graphs —cone⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([(0,1)],2)
 => 0
([],2)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 8
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(1,3),(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)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(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),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
 => 2

Description

The number of pairs of vertices of a graph with distance 2. This is the coefficient of the quadratic term of the Wiener polynomial.

Matching statistic: St000459
(load all 4 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => 0
([],2)
 => []
 => 0
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => 0
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => []
 => 0
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 6
([],5)
 => []
 => 0
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 8

Description

The hook length of the base cell of a partition. This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Matching statistic: St001311
(load all 5 compositions to match this statistic)

Mapping

Mp00203: Graphs —cone⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([(0,1)],2)
 => 0
([],2)
 => ([(0,2),(1,2)],3)
 => 0
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 0
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(1,3),(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)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(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),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
 => 8

Description

The cyclomatic number of a graph. This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as

$c + m - n$ , where

$c$ is the number of connected components,

$m$ is the number of edges and

$n$ is the number of vertices.

Matching statistic: St001341
(load all 2 compositions to match this statistic)

Mapping

Mp00259: Graphs —vertex addition⟶ Graphs
St001341: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],2)
 => 0
([],2)
 => ([],3)
 => 0
([(0,1)],2)
 => ([(1,2)],3)
 => 1
([],3)
 => ([],4)
 => 0
([(1,2)],3)
 => ([(2,3)],4)
 => 1
([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 2
([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => 3
([],4)
 => ([],5)
 => 0
([(2,3)],4)
 => ([(3,4)],5)
 => 1
([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => 2
([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2)],4)
 => ([(1,4),(2,3)],5)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
([],5)
 => ([],6)
 => 0
([(3,4)],5)
 => ([(4,5)],6)
 => 1
([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => 2
([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 4
([(1,4),(2,3)],5)
 => ([(2,5),(3,4)],6)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(2,5),(3,4),(4,5)],6)
 => 3
([(0,1),(2,4),(3,4)],5)
 => ([(1,2),(3,5),(4,5)],6)
 => 3
([(2,3),(2,4),(3,4)],5)
 => ([(3,4),(3,5),(4,5)],6)
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 8

Description

The number of edges in the center of a graph. The center of a graph is the set of vertices whose maximal distance to any other vertex is minimal. In particular, if the graph is disconnected, all vertices are in the certer.

Matching statistic: St001646
(load all 2 compositions to match this statistic)

Mapping

Mp00203: Graphs —cone⟶ Graphs
St001646: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([(0,1)],2)
 => 0
([],2)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 3
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 6
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 10
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 9
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 8
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 6
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 5
([(0,1),(0,4),(1,3),(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)
 => 4
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
([(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),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
 => 2

Description

The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on

$n$ vertices, which has maximal degree

$n-1$ and therefore has statistic

$\binom{n-1}{2}$ .

Matching statistic: St000450
(load all 5 compositions to match this statistic)

Mapping

Mp00203: Graphs —cone⟶ Graphs
St000450: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
([],2)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 7 = 6 + 1
([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 8 = 7 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7 = 6 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(0,4),(1,3),(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)
 => 7 = 6 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 8 = 7 + 1
([(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),(2,5),(3,4),(3,5),(4,5)],6)
 => 7 = 6 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7 = 6 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 8 = 7 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 9 = 8 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 8 = 7 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(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),(4,5)],6)
 => 9 = 8 + 1

Description

The number of edges minus the number of vertices plus 2 of a graph. When G is connected and planar, this is also the number of its faces. When

$G=(V,E)$ is a connected graph, this is its

$k$ -monochromatic index for

$k>2$ : for

$2\leq k\leq |V|$ , the

$k$ -monochromatic index of

$G$ is the maximum number of edge colors allowed such that for each set

$S$ of

$k$ vertices, there exists a monochromatic tree in

$G$ which contains all vertices from

$S$ . It is shown in [1] that for

$k>2$ , this is given by this statistic.

Matching statistic: St001034
(load all 9 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => []
 => []
 => 0
([],2)
 => []
 => []
 => 0
([(0,1)],2)
 => [1]
 => [1,0]
 => 1
([],3)
 => []
 => []
 => 0
([(1,2)],3)
 => [1]
 => [1,0]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,0,1,0]
 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([],4)
 => []
 => []
 => 0
([(2,3)],4)
 => [1]
 => [1,0]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => [1,0,1,0]
 => 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,2)],4)
 => [1,1]
 => [1,1,0,0]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([],5)
 => []
 => []
 => 0
([(3,4)],5)
 => [1]
 => [1,0]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => [1,0,1,0]
 => 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,4),(2,3)],5)
 => [1,1]
 => [1,1,0,0]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1,0,1,1,0,0]
 => 3
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 7
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 8

Description

The area of the parallelogram polyomino associated with the Dyck path. The (bivariate) generating function is given in [1].

Matching statistic: St000456
(load all 3 compositions to match this statistic)

Mapping

Mp00259: Graphs —vertex addition⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
([],2)
 => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
([(0,1)],2)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
([],3)
 => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
([(1,2)],3)
 => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
([],4)
 => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
([(2,3)],4)
 => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => ([(1,4),(2,3)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4 = 3 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 7 = 6 + 1
([],5)
 => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
([(3,4)],5)
 => ([(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => ([(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => ([(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(1,4),(2,3)],5)
 => ([(2,5),(3,4)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
 => ([(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 8 = 7 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 5 = 4 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => 7 = 6 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 6 = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 8 = 7 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 8 = 7 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => 9 = 8 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 8 = 7 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 9 = 8 + 1

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The following 69 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000293The number of inversions of a binary word. St001249Sum of the odd parts of a partition. St000290The major index of a binary word. St000395The sum of the heights of the peaks of a Dyck path. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001485The modular major index of a binary word. St001523The degree of symmetry of a Dyck path. St000171The degree of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000674The number of hills of a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000444The length of the maximal rise of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001161The major index north count of a Dyck path. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001645The pebbling number of a connected graph. St001725The harmonious chromatic number of a graph. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St001959The product of the heights of the peaks of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001622The number of join-irreducible elements of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St000438The position of the last up step in a Dyck path. St000981The length of the longest zigzag subpath. St001330The hat guessing number of a graph. St001644The dimension of a graph. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St000302The determinant of the distance matrix of a connected graph. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001060The distinguishing index of a graph.