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Identifier

St001798: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

([(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) => 0

([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 3

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

([],6) => 0

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 4

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

([(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) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

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

$F_{5} = 5 + 6\ q + 9\ q^{2} + 8\ q^{3} + 4\ q^{4} + 2\ q^{5}$

$F_{6} = 6 + 8\ q + 18\ q^{2} + 29\ q^{3} + 34\ q^{4} + 29\ q^{5} + 19\ q^{6} + 9\ q^{7} + 3\ q^{8} + q^{9}$

Description

The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph.
Let

$n(G)$ be the number of vertices of a graph

$G$ ,

$m(G)$ be its number of edges and let

$\alpha(G)$ be its independence number, St000093The cardinality of a maximal independent set of vertices of a graph.. Turán's theorem is that

$m(G) \geq m(T^c(n(G), \alpha(G)))$ where

$T^c(n, r)$ is the complement of the Turán graph.
This statistic records the difference.

References

[1] wikipedia:Turán's_theorem

Code

def statistic(G):
    T = graphs.TuranGraph(G.num_verts(), G.independent_set(value_only=True))
    return G.num_edges()-T.complement().num_edges()

Created

May 22, 2022 at 19:00 by Martin Rubey

Updated

May 22, 2022 at 19:00 by Martin Rubey