***************************************************************************** * 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: St001386 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- 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() ----------------------------------------------------------------------------- Statistic 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 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 24 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 40 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 12 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 44 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 48 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 26 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 20 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 36 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 4 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 5 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 20 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 16 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 52 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 14 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 6 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 14 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 16 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 13 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 6 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 10 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 28 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 10 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 12 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 12 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 8 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 11 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 10 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 3 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 2 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 2 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 4 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 12 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 8 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 28 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 12 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 6 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 1 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 8 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 2 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 4 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 0 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 1 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 0 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 6 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 0 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 4 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 0 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(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) => 0 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(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) => 0 ([(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) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 0 ([(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) => 0 ([(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) => 0 ([(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) => 0 ([(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) => 0 ----------------------------------------------------------------------------- Created: Apr 27, 2019 at 22:17 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 23, 2020 at 11:52 by Martin Rubey