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Identifier

St001802: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 4

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

([],3) => 27

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

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

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

([],4) => 256

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

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

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

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

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

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

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

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

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

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

([],5) => 3125

([(3,4)],5) => 250

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

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

([(0,4),(1,4),(2,4),(3,4)],5) => 260

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

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

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

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

([(0,4),(1,4),(2,3),(3,4)],5) => 60

([(1,4),(2,3),(2,4),(3,4)],5) => 70

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

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

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

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

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

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 180

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

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

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

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

([(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) => 10

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

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

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

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

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

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

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 64

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 60

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

([],6) => 46656

([(4,5)],6) => 2592

([(3,5),(4,5)],6) => 1296

([(2,5),(3,5),(4,5)],6) => 1080

([(1,5),(2,5),(3,5),(4,5)],6) => 1560

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 3130

([(2,5),(3,4)],6) => 576

([(2,5),(3,4),(4,5)],6) => 576

([(1,2),(3,5),(4,5)],6) => 288

([(3,4),(3,5),(4,5)],6) => 1296

([(1,5),(2,5),(3,4),(4,5)],6) => 360

([(0,1),(2,5),(3,5),(4,5)],6) => 256

([(2,5),(3,4),(3,5),(4,5)],6) => 504

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 374

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 288

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 282

([(2,4),(2,5),(3,4),(3,5)],6) => 1152

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 564

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 166

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 576

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 252

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 234

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 440

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 300

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 162

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 220

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 1080

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 316

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 680

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 396

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 192

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 282

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 1280

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 528

([(0,5),(1,4),(2,3)],6) => 216

([(1,5),(2,4),(3,4),(3,5)],6) => 252

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

([(1,2),(3,4),(3,5),(4,5)],6) => 288

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

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 192

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

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 104

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 192

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

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 60

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 296

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

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

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$F_{1} = q$

$F_{2} = q^{2} + q^{4}$

$F_{3} = 3\ q^{6} + q^{27}$

$F_{4} = q^{14} + 3\ q^{16} + 3\ q^{24} + q^{30} + 2\ q^{32} + q^{256}$

$F_{5} = q^{10} + q^{28} + 2\ q^{32} + q^{36} + 2\ q^{42} + q^{44} + 4\ q^{48} + q^{50} + q^{52} + 2\ q^{60} + q^{64} + q^{66} + q^{70} + q^{78} + 3\ q^{80} + q^{94} + 2\ q^{120} + 3\ q^{150} + q^{160} + q^{180} + q^{250} + q^{260} + q^{3125}$

Description

The number of endomorphisms of a graph.
An endomorphism of a graph

$(V, E)$ is a map

$f: V\to V$ such that for any edge

$(u,v)\in E$ also

$\big(f(u), f(v)\big)\in E$ .

Code

def statistic(G):
    G = G.relabel(inplace=False)
    n = G.num_verts()
    endomorphisms = 0
    for f in cartesian_product([list(range(n)) for _ in range(n)]):
        if all(G.has_edge(f[u], f[v]) for u, v in G.edges(labels=False)):
            endomorphisms += 1
    return endomorphisms

Created

Jun 06, 2022 at 18:38 by Martin Rubey

Updated

Jun 06, 2022 at 18:38 by Martin Rubey