***************************************************************************** * 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: St000343 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The number of spanning subgraphs of a graph. This is the number of subsets of the edge set of the graph, or the evaluation of the Tutte polynomial at $x=y=2$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(x): return 2^x.num_edges() def statistic_alternative(g): return g.tutte_polynomial().subs(x=Integer(2),y=Integer(2)) ----------------------------------------------------------------------------- Statistic values: ([],1) => 1 ([],2) => 1 ([(0,1)],2) => 2 ([],3) => 1 ([(1,2)],3) => 2 ([(0,2),(1,2)],3) => 4 ([(0,1),(0,2),(1,2)],3) => 8 ([],4) => 1 ([(2,3)],4) => 2 ([(1,3),(2,3)],4) => 4 ([(0,3),(1,3),(2,3)],4) => 8 ([(0,3),(1,2)],4) => 4 ([(0,3),(1,2),(2,3)],4) => 8 ([(1,2),(1,3),(2,3)],4) => 8 ([(0,3),(1,2),(1,3),(2,3)],4) => 16 ([(0,2),(0,3),(1,2),(1,3)],4) => 16 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 32 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 64 ([],5) => 1 ([(3,4)],5) => 2 ([(2,4),(3,4)],5) => 4 ([(1,4),(2,4),(3,4)],5) => 8 ([(0,4),(1,4),(2,4),(3,4)],5) => 16 ([(1,4),(2,3)],5) => 4 ([(1,4),(2,3),(3,4)],5) => 8 ([(0,1),(2,4),(3,4)],5) => 8 ([(2,3),(2,4),(3,4)],5) => 8 ([(0,4),(1,4),(2,3),(3,4)],5) => 16 ([(1,4),(2,3),(2,4),(3,4)],5) => 16 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 32 ([(1,3),(1,4),(2,3),(2,4)],5) => 16 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 32 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 32 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 32 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 64 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 64 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 128 ([(0,4),(1,3),(2,3),(2,4)],5) => 16 ([(0,1),(2,3),(2,4),(3,4)],5) => 16 ([(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) => 64 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 32 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 64 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 128 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 64 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 64 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 128 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 256 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 128 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 256 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 512 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1024 ([],6) => 1 ([(4,5)],6) => 2 ([(3,5),(4,5)],6) => 4 ([(2,5),(3,5),(4,5)],6) => 8 ([(1,5),(2,5),(3,5),(4,5)],6) => 16 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 32 ([(2,5),(3,4)],6) => 4 ([(2,5),(3,4),(4,5)],6) => 8 ([(1,2),(3,5),(4,5)],6) => 8 ([(3,4),(3,5),(4,5)],6) => 8 ([(1,5),(2,5),(3,4),(4,5)],6) => 16 ([(0,1),(2,5),(3,5),(4,5)],6) => 16 ([(2,5),(3,4),(3,5),(4,5)],6) => 16 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 32 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 32 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 64 ([(2,4),(2,5),(3,4),(3,5)],6) => 16 ([(0,5),(1,5),(2,4),(3,4)],6) => 16 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 32 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 32 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 32 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 32 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 32 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 64 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 64 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 64 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 256 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,5),(1,4),(2,3)],6) => 8 ([(1,5),(2,4),(3,4),(3,5)],6) => 16 ([(0,1),(2,5),(3,4),(4,5)],6) => 16 ([(1,2),(3,4),(3,5),(4,5)],6) => 16 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 32 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 32 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 32 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 64 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 64 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 128 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 32 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 64 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 64 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 32 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 32 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 64 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 64 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 64 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 64 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 64 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => 128 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => 128 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 64 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 128 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 256 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 256 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 256 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 512 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 128 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 256 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 256 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 64 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 128 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 128 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 128 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 128 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => 256 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 256 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 512 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 512 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 1024 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => 2048 ([(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) => 4096 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => 64 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => 128 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => 128 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 128 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 256 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 256 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => 256 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 512 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 512 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 1024 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 256 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 256 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 512 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1024 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 512 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1024 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(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) => 4096 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1024 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(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) => 4096 ([(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) => 8192 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2048 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 2048 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2048 ([(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) => 4096 ([(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) => 4096 ([(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) => 8192 ([(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) => 16384 ([(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) => 32768 ----------------------------------------------------------------------------- Created: Dec 23, 2015 at 09:06 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 23, 2015 at 09:06 by Martin Rubey