*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* 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