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Identifier

St000465: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

([(0,3),(1,2),(2,3)],4) => 10

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

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

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 26

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

([],5) => 0

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

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

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

([(0,4),(1,4),(2,4),(3,4)],5) => 20

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

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

([(0,1),(2,4),(3,4)],5) => 8

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

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

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

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 26

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

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 26

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 34

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 30

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

([(0,4),(1,3),(2,3),(2,4)],5) => 14

([(0,1),(2,3),(2,4),(3,4)],5) => 14

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

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

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 20

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 30

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

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

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

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

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

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 40

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

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

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

([],6) => 0

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

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

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

([(1,5),(2,5),(3,5),(4,5)],6) => 20

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 30

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

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

([(1,2),(3,5),(4,5)],6) => 8

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

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

([(0,1),(2,5),(3,5),(4,5)],6) => 14

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

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 24

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 26

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 36

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 22

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 20

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 26

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 24

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 22

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 30

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 34

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

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 44

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 30

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 28

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 38

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 44

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 54

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 48

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 66

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

([(1,5),(2,4),(3,4),(3,5)],6) => 14

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

([(1,2),(3,4),(3,5),(4,5)],6) => 14

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 20

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 22

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 20

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 30

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

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

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 20

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 28

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 30

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 26

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Description

The first Zagreb index of a graph.
This is the sum of the squares of the degrees of the vertices,
$$\sum_{v \in V(G)} d^2(v) = \sum_{\{u,v\}\in E(G)} \big(d(u)+d(v)\big)$$
where $d(u)$ is the degree of the vertex $u$.

References

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

Code

def statistic(G):
    return sum(d^2 for d in G.degree_iterator())

def statistic_alt(G):
    return sum( G.degree(v) + G.degree(w) for v,w in G.edges(labels=False) )

Created

Apr 07, 2016 at 21:14 by Martin Rubey

Updated

Aug 10, 2016 at 15:28 by Christian Stump