Identifier
Values
=>
Cc0020;cc-rep
[1]=>([],1)=>0 [1,1]=>([(0,1)],2)=>1 [2]=>([],2)=>1 [1,1,1]=>([(0,1),(0,2),(1,2)],3)=>2 [1,2]=>([(1,2)],3)=>1 [2,1]=>([(0,2),(1,2)],3)=>1 [3]=>([],3)=>2 [1,1,1,1]=>([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>3 [1,1,2]=>([(1,2),(1,3),(2,3)],4)=>2 [1,2,1]=>([(0,3),(1,2),(1,3),(2,3)],4)=>1 [1,3]=>([(2,3)],4)=>2 [2,1,1]=>([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>2 [2,2]=>([(1,3),(2,3)],4)=>1 [3,1]=>([(0,3),(1,3),(2,3)],4)=>2 [4]=>([],4)=>3 [1,1,1,1,1]=>([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4 [1,1,1,2]=>([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3 [1,1,2,1]=>([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>2 [1,1,3]=>([(2,3),(2,4),(3,4)],5)=>3 [1,2,1,1]=>([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>2 [1,2,2]=>([(1,4),(2,3),(2,4),(3,4)],5)=>1 [1,3,1]=>([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>2 [1,4]=>([(3,4)],5)=>3 [2,1,1,1]=>([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3 [2,1,2]=>([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>2 [2,2,1]=>([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>1 [2,3]=>([(2,4),(3,4)],5)=>2 [3,1,1]=>([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3 [3,2]=>([(1,4),(2,4),(3,4)],5)=>2 [4,1]=>([(0,4),(1,4),(2,4),(3,4)],5)=>3 [5]=>([],5)=>4 [1,1,1,1,1,1]=>([(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)=>5 [1,1,1,1,2]=>([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 [1,1,1,2,1]=>([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [1,1,1,3]=>([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 [1,1,2,1,1]=>([(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)=>3 [1,1,2,2]=>([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [1,1,3,1]=>([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [1,1,4]=>([(3,4),(3,5),(4,5)],6)=>4 [1,2,1,1,1]=>([(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)=>3 [1,2,1,2]=>([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [1,2,2,1]=>([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 [1,2,3]=>([(2,5),(3,4),(3,5),(4,5)],6)=>2 [1,3,1,1]=>([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [1,3,2]=>([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2 [1,4,1]=>([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>3 [1,5]=>([(4,5)],6)=>4 [2,1,1,1,1]=>([(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)=>4 [2,1,1,2]=>([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [2,1,2,1]=>([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [2,1,3]=>([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [2,2,1,1]=>([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [2,2,2]=>([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 [2,3,1]=>([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [2,4]=>([(3,5),(4,5)],6)=>3 [3,1,1,1]=>([(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)=>4 [3,1,2]=>([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 [3,2,1]=>([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 [3,3]=>([(2,5),(3,5),(4,5)],6)=>3 [4,1,1]=>([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 [4,2]=>([(1,5),(2,5),(3,5),(4,5)],6)=>3 [5,1]=>([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>4 [6]=>([],6)=>5 [1,1,1,1,1,1,1]=>([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>6 [1,1,1,1,1,2]=>([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [1,1,1,1,2,1]=>([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,1,1,1,3]=>([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [1,1,1,2,1,1]=>([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,1,1,2,2]=>([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,1,1,3,1]=>([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,1,1,4]=>([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [1,1,2,1,1,1]=>([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,1,2,1,2]=>([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,1,2,2,1]=>([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [1,1,2,3]=>([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,1,3,2]=>([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,1,4,1]=>([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,1,5]=>([(4,5),(4,6),(5,6)],7)=>5 [1,2,1,1,1,1]=>([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,2,1,1,2]=>([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,2,1,2,1]=>([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [1,2,1,3]=>([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,2,2,1,1]=>([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [1,2,2,2]=>([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>1 [1,2,3,1]=>([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [1,2,4]=>([(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,3,1,1,1]=>([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,3,1,2]=>([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,3,2,1]=>([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [1,3,3]=>([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,4,1,1]=>([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,4,2]=>([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>3 [1,5,1]=>([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>4 [1,6]=>([(5,6)],7)=>5 [2,1,1,1,1,1]=>([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [2,1,1,1,2]=>([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [2,1,1,2,1]=>([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,1,1,3]=>([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [2,1,2,1,1]=>([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,1,2,2]=>([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [2,1,3,1]=>([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,1,4]=>([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [2,2,1,1,1]=>([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,2,1,2]=>([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [2,2,2,1]=>([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>1 [2,2,3]=>([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [2,3,1,1]=>([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,3,2]=>([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [2,4,1]=>([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [2,5]=>([(4,6),(5,6)],7)=>4 [3,1,1,1,1]=>([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [3,1,1,2]=>([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [3,1,2,1]=>([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [3,1,3]=>([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [3,2,1,1]=>([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [3,2,2]=>([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2 [3,3,1]=>([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [3,4]=>([(3,6),(4,6),(5,6)],7)=>4 [4,1,1,1]=>([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [4,1,2]=>([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4 [4,2,1]=>([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3 [4,3]=>([(2,6),(3,6),(4,6),(5,6)],7)=>4 [5,1,1]=>([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>5 [5,2]=>([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>4 [6,1]=>([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>5 [7]=>([],7)=>6
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Description
The rigidity index of a graph.
A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter.
This statistic yields the minimal size of a base for the automorphism group of a graph.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.