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Identifier

St000464: Graphs ⟶ ℤ

Values

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

([(0,2),(1,2)],3) => 10

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

([(0,3),(1,3),(2,3)],4) => 24

([(0,3),(1,2),(2,3)],4) => 28

([(0,3),(1,2),(1,3),(2,3)],4) => 30

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 34

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 36

([(0,4),(1,4),(2,4),(3,4)],5) => 44

([(0,4),(1,4),(2,3),(3,4)],5) => 52

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 54

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 61

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 58

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 62

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 66

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 68

([(0,4),(1,3),(2,3),(2,4)],5) => 60

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 64

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 64

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 60

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 66

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 70

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 67

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 68

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 74

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 72

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 76

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 78

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 80

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 70

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 82

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 84

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 98

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 86

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 96

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 92

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 104

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 106

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 100

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 106

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 112

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 114

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 94

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 98

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 98

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 100

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 100

([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 96

([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 105

([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 105

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 108

([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 110

([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 110

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 118

([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 106

([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 104

([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 112

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 112

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 108

([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 110

([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 106

([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 106

([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 106

([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 114

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 116

([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 113

([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 112

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 118

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 116

([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 110

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 116

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 122

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 124

([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 110

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 115

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 118

([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 108

([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 114

([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 110

([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 110

([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 114

([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 111

([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 118

([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 119

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 114

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 120

([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 120

([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 122

([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 124

([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 126

([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128

([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 124

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 121

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 124

([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 122

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 120

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 126

([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 132

([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 126

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Description

The Schultz index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].

References

[1] http://www.mat.uc.pt/~combdays/3rdAbstracts/PaulaRama3rdcombday.pdf
[2] Klein, D. J., Mihalic, Z., Plavsic, D., Trinajstic, N. Molecular topological index: a relation with the Wiener index DOI:10.1021/ci00008a008

Code

def statistic(G):
    """
    sage: all((statistic(G)==4*G.wiener_index()-n*(n-1)) for n in range(2,7) for G in graphs(n) if G.is_tree())
    """
    return sum((G.degree(u)+G.degree(v))*G.distance(u, v) for (u,v) in Subsets(G.vertices(), 2))

Created

Apr 07, 2016 at 16:29 by Martin Rubey

Updated

Apr 07, 2016 at 16:29 by Martin Rubey