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Identifier

St001521: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

([(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) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 3

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

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

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

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

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

([(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) => 5

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

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

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

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

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

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

([(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) => 4

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

([(2,5),(3,5),(4,5)],6) => 9

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

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 10

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

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

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

([(3,4),(3,5),(4,5)],6) => 9

([(1,5),(2,5),(3,4),(4,5)],6) => 9

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

([(2,5),(3,4),(3,5),(4,5)],6) => 11

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 9

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

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

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 9

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

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

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 8

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 9

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 9

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 8

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 9

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 13

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 8

([(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) => 7

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

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

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

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 9

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

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 9

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

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 10

([(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) => 8

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 9

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

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Description

Half the total irregularity of a graph.
This is half the sum of the absolute values of the degree differences of all pairs of vertices:
$$ \frac{1}{4}\sum_{u,v} |d_u-d_v| $$
It is easy to show by induction on the number of edges that this is an integer.

References

[1] Abdo, H., Brandt, S., Dimitrov, D. The total irregularity of a graph MathSciNet:3214266

Code

def statistic(G):
    return sum(abs(G.degree(u)-G.degree(v)) for u in range(G.num_verts()) for v in range(u))/2

Created

Feb 29, 2020 at 18:01 by Martin Rubey

Updated

Feb 29, 2020 at 18:01 by Martin Rubey