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Identifier

St001459: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

([(0,1)],2) => 2

([],3) => 0

([(1,2)],3) => 2

([(0,2),(1,2)],3) => 1

([(0,1),(0,2),(1,2)],3) => 3

([],4) => 0

([(2,3)],4) => 2

([(1,3),(2,3)],4) => 1

([(0,3),(1,3),(2,3)],4) => 1

([(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) => 4

([(0,2),(0,3),(1,2),(1,3)],4) => 0

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4

([],5) => 0

([(3,4)],5) => 2

([(2,4),(3,4)],5) => 1

([(1,4),(2,4),(3,4)],5) => 1

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

([(1,4),(2,3)],5) => 4

([(1,4),(2,3),(3,4)],5) => 4

([(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) => 3

([(1,4),(2,3),(2,4),(3,4)],5) => 4

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

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

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

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

([(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) => 0

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

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

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

([(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) => 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) => 1

([(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) => 5

([(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) => 5

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

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

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

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

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

([],6) => 0

([(4,5)],6) => 2

([(3,5),(4,5)],6) => 1

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

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

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

([(2,5),(3,4)],6) => 4

([(2,5),(3,4),(4,5)],6) => 4

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

([(3,4),(3,5),(4,5)],6) => 3

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

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

([(2,5),(3,4),(3,5),(4,5)],6) => 4

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

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

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

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

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

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

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

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q + q^{2} + q^{3}$

$F_{4} = 2 + 2\ q + 2\ q^{2} + q^{3} + 4\ q^{4}$

$F_{5} = 3 + 5\ q + 5\ q^{2} + 8\ q^{3} + 4\ q^{4} + 9\ q^{5}$

$F_{6} = 8 + 12\ q + 25\ q^{2} + 18\ q^{3} + 27\ q^{4} + 9\ q^{5} + 57\ q^{6}$

Description

The number of zero columns in the nullspace of a graph.

Code

def statistic(G):
    n = G.num_verts()    
    K = G.adjacency_matrix().kernel().matrix()
    return sum(1 for i in range(n) if all(row[i] == 0 for row in K))

Created

Aug 03, 2019 at 21:34 by Martin Rubey

Updated

Aug 03, 2019 at 21:34 by Martin Rubey