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Identifier

St000466: Graphs ⟶ ℤ

Values

([],1) => 0

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

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

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

([(0,3),(1,3),(2,3)],4) => 15

([(0,3),(1,2),(2,3)],4) => 19

([(0,3),(1,2),(1,3),(2,3)],4) => 27

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 41

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

([(0,4),(1,4),(2,4),(3,4)],5) => 28

([(0,4),(1,4),(2,3),(3,4)],5) => 36

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 46

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 57

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 50

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

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 88

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

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

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

([(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) => 77

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 93

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

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 85

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

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 99

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 138

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 160

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 45

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 57

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 69

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 73

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 61

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

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 77

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 95

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

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 99

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 123

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 144

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 153

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 69

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 85

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 101

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 90

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 95

([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 81

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

([(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) => 128

([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 85

([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 102

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 109

([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 93

([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 93

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

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

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

([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 135

([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 107

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

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

([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 147

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 135

([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 134

([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 140

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 163

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 142

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

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 154

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 177

([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 190

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

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 140

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 161

([(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) => 129

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

([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 125

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

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

([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 149

([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 142

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 146

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 169

([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 156

([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 150

([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 182

([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 178

([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 203

([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 169

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 160

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 185

([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 175

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 168

([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 195

([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 227

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Description

The Gutman (or modified 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 modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].

References

[1] http://www.mat.uc.pt/~combdays/3rdAbstracts/PaulaRama3rdcombday.pdf

Code

def statistic(G):
    """
    sage: all((statistic(G)==4*G.wiener_index()-(n-1)*(2*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 21:09 by Martin Rubey

Updated

Apr 07, 2016 at 21:09 by Martin Rubey