Values
=>
Cc0020;cc-rep
([],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
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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.
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
Code
def statistic(g):
return g.num_verts() - g.num_edges()
Created
Jul 27, 2022 at 13:04 by Harry Richman
Updated
Jul 27, 2022 at 13:04 by Harry Richman
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