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Identifier

St001386: Graphs ⟶ ℤ

Values

([],0) => 1

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

([],4) => 1

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

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

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

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 15

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

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 15

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 6

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

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

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

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

([],6) => 1

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

([(3,5),(4,5)],6) => 33

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

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

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

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

([(2,5),(3,4),(4,5)],6) => 68

([(1,2),(3,5),(4,5)],6) => 59

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

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

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

([(2,5),(3,4),(3,5),(4,5)],6) => 52

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

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 41

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

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

([(0,5),(1,5),(2,4),(3,4)],6) => 17

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 62

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 52

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

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 68

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 16

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

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 52

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 50

([(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) => 7

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 18

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

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 18

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

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

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

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

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

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

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

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

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

([(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) => 36

([(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) => 8

([(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) => 32

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

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 2\ q + 2\ q^{3}$

$F_{4} = 1 + 3\ q + 3\ q^{2} + q^{4} + q^{5} + q^{6} + q^{8}$

$F_{5} = 1 + 2\ q + q^{2} + 3\ q^{3} + q^{4} + 5\ q^{6} + q^{7} + 3\ q^{9} + q^{12} + q^{14} + 2\ q^{15} + q^{18} + q^{21} + 4\ q^{24} + 3\ q^{30} + 3\ q^{36} + q^{42}$

$F_{6} = 42 + 8\ q + 11\ q^{2} + q^{3} + 9\ q^{4} + q^{5} + 10\ q^{6} + q^{7} + 8\ q^{8} + 8\ q^{10} + 3\ q^{11} + 8\ q^{12} + q^{13} + 3\ q^{14} + 3\ q^{16} + q^{17} + 3\ q^{18} + 3\ q^{20} + 2\ q^{22} + q^{23} + q^{24} + 3\ q^{26} + 2\ q^{28} + 2\ q^{32} + q^{33} + 3\ q^{36} + q^{40} + q^{41} + q^{44} + 2\ q^{48} + q^{50} + 4\ q^{52} + q^{54} + q^{59} + q^{62} + 2\ q^{68} + q^{82} + q^{98}$

Description

The number of prime labellings of a graph.
A prime labelling of a graph is a bijective labelling of the vertices with the numbers

$\{1,\dots, |V(G)|\}$ such that adjacent vertices have coprime labels.

References

[1] Andreson, M. Prime labelling of graphs MathOverflow:191182

Code

def statistic(G):
    G.relabel(inplace=False)
    n = G.num_verts()
    good = 0
    for pi in Permutations(n):
        if all(gcd(pi[u], pi[v]) == 1 for u, v in G.edges(labels=False)):
            good += 1
    return good/G.automorphism_group().cardinality()

Created

Apr 27, 2019 at 22:17 by Martin Rubey

Updated

Dec 23, 2020 at 11:52 by Martin Rubey