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Identifier
Values
=>
Cc0020;cc-rep
([],0)=>1 ([],1)=>2 ([],2)=>4 ([(0,1)],2)=>4 ([],3)=>8 ([(1,2)],3)=>8 ([(0,2),(1,2)],3)=>7 ([(0,1),(0,2),(1,2)],3)=>5 ([],4)=>16 ([(2,3)],4)=>16 ([(1,3),(2,3)],4)=>14 ([(0,3),(1,3),(2,3)],4)=>12 ([(0,3),(1,2)],4)=>16 ([(0,3),(1,2),(2,3)],4)=>12 ([(1,2),(1,3),(2,3)],4)=>10 ([(0,3),(1,2),(1,3),(2,3)],4)=>8 ([(0,2),(0,3),(1,2),(1,3)],4)=>10 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>6 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>6 ([],5)=>32 ([(3,4)],5)=>32 ([(2,4),(3,4)],5)=>28 ([(1,4),(2,4),(3,4)],5)=>24 ([(0,4),(1,4),(2,4),(3,4)],5)=>21 ([(1,4),(2,3)],5)=>32 ([(1,4),(2,3),(3,4)],5)=>24 ([(0,1),(2,4),(3,4)],5)=>28 ([(2,3),(2,4),(3,4)],5)=>20 ([(0,4),(1,4),(2,3),(3,4)],5)=>20 ([(1,4),(2,3),(2,4),(3,4)],5)=>16 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>13 ([(1,3),(1,4),(2,3),(2,4)],5)=>20 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>16 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>12 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>13 ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>9 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)=>13 ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>7 ([(0,4),(1,3),(2,3),(2,4)],5)=>21 ([(0,1),(2,3),(2,4),(3,4)],5)=>20 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>14 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)=>9 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>17 ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>11 ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)=>7 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>10 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>12 ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>9 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>7 ([(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)=>7 ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>7 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>7 ([],6)=>64 ([(4,5)],6)=>64 ([(3,5),(4,5)],6)=>56 ([(2,5),(3,5),(4,5)],6)=>48 ([(1,5),(2,5),(3,5),(4,5)],6)=>42 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>38 ([(2,5),(3,4)],6)=>64 ([(2,5),(3,4),(4,5)],6)=>48 ([(1,2),(3,5),(4,5)],6)=>56 ([(3,4),(3,5),(4,5)],6)=>40 ([(1,5),(2,5),(3,4),(4,5)],6)=>40 ([(0,1),(2,5),(3,5),(4,5)],6)=>48 ([(2,5),(3,4),(3,5),(4,5)],6)=>32 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>34 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>26 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>22 ([(2,4),(2,5),(3,4),(3,5)],6)=>40 ([(0,5),(1,5),(2,4),(3,4)],6)=>49 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>32 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>35 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>26 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>33 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>26 ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18 ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>21 ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>26 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>26 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>20 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(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)=>16 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,4),(2,3)],6)=>64 ([(1,5),(2,4),(3,4),(3,5)],6)=>42 ([(0,1),(2,5),(3,4),(4,5)],6)=>48 ([(1,2),(3,4),(3,5),(4,5)],6)=>40 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>34 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>28 ([(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)=>22 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>18 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>14 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>34 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>25 ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>22 ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)=>28 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>20 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>22 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>17 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>15 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>37 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>40 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>35 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>28 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>25 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>23 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>23 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>18 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>16 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>15 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>16 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>22 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)=>14 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>11 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,4),(0,5),(1,4),(1,5),(2,3),(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)],6)=>14 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>12 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>20 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>18 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>13 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>14 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>29 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)=>21 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>18 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>19 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>18 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>18 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>13 ([(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)=>12 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>8 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)=>17 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>16 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>12 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>11 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>11 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>11 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>8 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>17 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>11 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>10 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>8 ([(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)=>8 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>25 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>18 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>17 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>12 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>16 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)=>13 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>10 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>9 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)=>15 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>14 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>10 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>13 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>9 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>8 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(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)=>8 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>10 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>9 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>11 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>14 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>10 ([(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 ([(0,3),(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 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>8 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>10 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>9 ([(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)=>8 ([(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)=>8 ([(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)=>8 ([(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)=>8
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Description
The number of closed sets in a graph.
A subset $S$ of the set of vertices is a closed set, if for any pair of distinct elements of $S$ the intersection of the corresponding neighbourhoods is a subset of $S$:
$$ \forall a, b\in S: N(a)\cap N(b) \subseteq S. $$
References
[1] Koh, K. M., Poh, K. S. On the spectrum of the closed-set lattice of a graph MathSciNet:1165806
Code
def is_closed_set(G, S):
    for u, v in Subsets(S, 2):
        if not set(G[u]).intersection(G[v]).issubset(S):
            return False
    return True

def statistic(G):
    return sum(1 for S in Subsets(G.vertices()) if is_closed_set(G, S))
Created
Mar 31, 2021 at 21:51 by Martin Rubey
Updated
Mar 31, 2021 at 21:51 by Martin Rubey