***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001802 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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 $\big(f(u), f(v)\big)\in E$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------- Statistic 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 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 264 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 162 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 99 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 153 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 216 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 136 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 104 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 340 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 108 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 234 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 74 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 101 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 192 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 86 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 76 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 130 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 86 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 88 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 864 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 152 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 468 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 312 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 24 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 84 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 174 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 119 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 112 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 114 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 164 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 312 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 292 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 198 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 200 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 144 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 522 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 288 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 174 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 384 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 132 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 362 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 96 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 44 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 144 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 75 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 99 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 113 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 58 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 82 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 768 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 144 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 120 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 180 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 168 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 220 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 360 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 194 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 244 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 117 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 84 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 130 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 168 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 232 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 144 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 120 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 56 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 336 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 72 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 246 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 156 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 40 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 70 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 180 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 74 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 196 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 110 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 112 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 52 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 52 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 50 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 10 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 36 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 70 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 124 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 90 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 124 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 152 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 150 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 228 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 720 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 504 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 372 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 176 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 240 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 384 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 208 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 288 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 720 ----------------------------------------------------------------------------- Created: Jun 06, 2022 at 18:38 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 06, 2022 at 18:38 by Martin Rubey