Identifier
Values
[1] => 10 => [1,2] => ([(1,2)],3) => 0
[2] => 100 => [1,3] => ([(2,3)],4) => 0
[1,1] => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => 0
[3] => 1000 => [1,4] => ([(3,4)],5) => 0
[2,1] => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => 1
[1,1,1] => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
[4] => 10000 => [1,5] => ([(4,5)],6) => 0
[3,1] => 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 1
[2,2] => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => 0
[2,1,1] => 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[1,1,1,1] => 11110 => [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) => 0
[3,2] => 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => 1
[2,2,1] => 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[3,3] => 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => 0
[2,2,2] => 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 0
[] => => [1] => ([],1) => 0
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Description
The minimal number of edges to add or remove to make a graph edge transitive.
A graph is edge transitive, if for any two edges, there is an automorphism that maps one edge to the other.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
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.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.