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Identifier

St001374: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

([(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) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 9

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

([(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) => 6

([(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) => 2

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

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

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

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

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

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

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

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

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

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

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

([(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) => 21

([(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) => 16

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

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

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

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

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

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

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 21

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

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

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

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

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

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 23

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

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

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

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

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

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

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

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

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

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 21

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

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

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

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

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

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 19

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Description

The Padmakar-Ivan index of a graph.
For an edge $e=(u, v)$, let $n_{e, u}$ be the number of edges in a graph $G$ induced by the set of vertices $\{w: d(u, w) < d(v, w)\}$, where $d(u,v)$ denotes the distance between $u$ and $v$.
Then the PI-index of $G$ is
$$\sum_{e=(u,v)} n_{e, u} + n_{e, v}.$$

References

[1] wikipedia:Padmakar-Ivan_index

Code

def vertices_closer_to_first(u, v, G):
    return [w for w in G.vertices() if G.distance(w, u) < G.distance(w, v)]

def statistic(G):
    return sum(G.subgraph(vertices_closer_to_first(u, v, G)).num_edges() + G.subgraph(vertices_closer_to_first(v, u, G)).num_edges()
               for u, v in G.edges(labels=False))

Created

Mar 26, 2019 at 23:02 by Martin Rubey

Updated

Dec 23, 2020 at 11:51 by Martin Rubey