***************************************************************************** * 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: St001828 ----------------------------------------------------------------------------- Collection: Graphs ----------------------------------------------------------------------------- Description: The Euler characteristic of a graph. The '''Euler characteristic''' $\chi$ of a topological space is the alternating sum of the dimensions of the homology groups $$\chi(X) = \sum_{k \geq 0} (-1)^k \dim H_k(X).$$ For a finite simplicial complex, this is equal to the alternating sum $\sum_{k\geq 0} (-1)^k f_k$ where $f_k$ the number of $k$-dimensional simplices. A (simple) graph is a simplicial complex of dimension at most one; its vertices are the 0-simplices and its edges are the 1-simplices. For a connected graph, the Euler characteristic is equal to $1 - g$ where $g$ is the cyclomatic number. ----------------------------------------------------------------------------- References: [1] [[wikipedia:Euler_characteristic]] [2] The cyclomatic number of a graph. [[St001311]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: ([],1) => 1 ([],2) => 2 ([(0,1)],2) => 1 ([],3) => 3 ([(1,2)],3) => 2 ([(0,2),(1,2)],3) => 1 ([(0,1),(0,2),(1,2)],3) => 0 ([],4) => 4 ([(2,3)],4) => 3 ([(1,3),(2,3)],4) => 2 ([(0,3),(1,3),(2,3)],4) => 1 ([(0,3),(1,2)],4) => 2 ([(0,3),(1,2),(2,3)],4) => 1 ([(1,2),(1,3),(2,3)],4) => 1 ([(0,3),(1,2),(1,3),(2,3)],4) => 0 ([(0,2),(0,3),(1,2),(1,3)],4) => 0 ([(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) => -2 ([],5) => 5 ([(3,4)],5) => 4 ([(2,4),(3,4)],5) => 3 ([(1,4),(2,4),(3,4)],5) => 2 ([(0,4),(1,4),(2,4),(3,4)],5) => 1 ([(1,4),(2,3)],5) => 3 ([(1,4),(2,3),(3,4)],5) => 2 ([(0,1),(2,4),(3,4)],5) => 2 ([(2,3),(2,4),(3,4)],5) => 2 ([(0,4),(1,4),(2,3),(3,4)],5) => 1 ([(1,4),(2,3),(2,4),(3,4)],5) => 1 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0 ([(1,3),(1,4),(2,3),(2,4)],5) => 1 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -1 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => -1 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -2 ([(0,4),(1,3),(2,3),(2,4)],5) => 1 ([(0,1),(2,3),(2,4),(3,4)],5) => 1 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 0 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => -1 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 0 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => -1 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => -2 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => -1 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -1 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -2 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -3 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => -2 ([(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) => -4 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => -5 ([],6) => 6 ([(4,5)],6) => 5 ([(3,5),(4,5)],6) => 4 ([(2,5),(3,5),(4,5)],6) => 3 ([(1,5),(2,5),(3,5),(4,5)],6) => 2 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 1 ([(2,5),(3,4)],6) => 4 ([(2,5),(3,4),(4,5)],6) => 3 ([(1,2),(3,5),(4,5)],6) => 3 ([(3,4),(3,5),(4,5)],6) => 3 ([(1,5),(2,5),(3,4),(4,5)],6) => 2 ([(0,1),(2,5),(3,5),(4,5)],6) => 2 ([(2,5),(3,4),(3,5),(4,5)],6) => 2 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(2,4),(2,5),(3,4),(3,5)],6) => 2 ([(0,5),(1,5),(2,4),(3,4)],6) => 2 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 1 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 1 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 0 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => -1 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => -2 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,5),(1,4),(2,3)],6) => 3 ([(1,5),(2,4),(3,4),(3,5)],6) => 2 ([(0,1),(2,5),(3,4),(4,5)],6) => 2 ([(1,2),(3,4),(3,5),(4,5)],6) => 2 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 1 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 1 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => -1 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 1 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 0 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 0 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => 1 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => 1 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => 0 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 0 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => -1 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => -1 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -1 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => -1 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => -2 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -2 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -2 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -3 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => -1 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -1 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -2 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -2 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 0 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => -1 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => -1 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => -1 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -1 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => -1 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -2 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => -2 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => -2 ([(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) => -3 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -3 ([(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) => -5 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => -3 ([(0,1),(0,2),(0,3),(1,4),(1,5),(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)],6) => -4 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => -5 ([(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) => -6 ([(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) => -1 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => -1 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -1 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -2 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -2 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => -2 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => -3 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -3 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => -4 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -5 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => -2 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -2 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => -3 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => -4 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => -3 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -4 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -5 ([(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) => -6 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => -4 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -5 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -4 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -5 ([(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) => -6 ([(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) => -7 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -5 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => -5 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => -5 ([(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) => -6 ([(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) => -6 ([(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) => -7 ([(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) => -8 ([(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) => -9 ----------------------------------------------------------------------------- Created: Jul 27, 2022 at 13:04 by Harry Richman ----------------------------------------------------------------------------- Last Updated: Jul 27, 2022 at 13:04 by Harry Richman