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Identifier

St000403: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

([(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) => 0

([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 12

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

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

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 22

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

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

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

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

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

([(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) => 10

([(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) => 4

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

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

([(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) => 16

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 12

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

([(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) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 12

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

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

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

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

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 16

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

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

([(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) => 22

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 18

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 32

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

([(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) => 6

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 42

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12

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

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

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

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

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

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

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

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

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

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

([(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) => 17

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 10

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

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

$F_{5} = 20 + 2\ q^{2} + 3\ q^{4} + 2\ q^{5} + q^{6} + q^{8} + q^{10} + 2\ q^{12} + q^{16} + q^{22}$

$F_{6} = 50 + 7\ q^{2} + 6\ q^{4} + 3\ q^{5} + 8\ q^{6} + 2\ q^{7} + 10\ q^{8} + 2\ q^{9} + 7\ q^{10} + 2\ q^{11} + 9\ q^{12} + q^{13} + 2\ q^{14} + 3\ q^{15} + 5\ q^{16} + 2\ q^{17} + 6\ q^{18} + q^{19} + 2\ q^{20} + 2\ q^{21} + 2\ q^{22} + q^{23} + 3\ q^{24} + q^{26} + 2\ q^{27} + 2\ q^{29} + 5\ q^{30} + 3\ q^{32} + q^{33} + q^{34} + q^{36} + q^{42} + q^{44} + q^{45} + q^{60}$

Description

The Szeged index minus the Wiener index of a graph.
It is known that the Szeged index is at least as much as the Wiener index. For

$2$ -connected graphs on

$n$ vertices, the difference is at least

$2n-6$ .
For the two individual statistics see St000263The Szeged index of a graph. and St000265The Wiener index of a graph..

References

[1] Bonamy, M., Knor, M., Lužar, B., Pinlou, A., Škrekovski, R. On the difference between the Szeged and Wiener index arXiv:1602.05184
[2] Klavžar, S., Nadjafi-Arani, M. J. Improved bounds on the difference between the Szeged index and the Wiener index of graphs MathSciNet:3168520
[3] Klavžar, S., Rajapakse, A., Gutman, I. The Szeged and the Wiener index of graphs MathSciNet:1415471

Code

def statistic(G):
    return G.szeged_index() - G.wiener_index()

Created

Feb 23, 2016 at 20:41 by Christian Stump

Updated

Feb 23, 2016 at 21:36 by Christian Stump