Identifier
Values
[1] => ([],1) => [] => 1
[1,1] => ([(0,1)],2) => [1] => 1
[2] => ([],2) => [] => 1
[1,1,1] => ([(0,1),(0,2),(1,2)],3) => [3] => 2
[1,2] => ([(1,2)],3) => [1] => 1
[2,1] => ([(0,2),(1,2)],3) => [2] => 2
[3] => ([],3) => [] => 1
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [6] => 8
[1,1,2] => ([(1,2),(1,3),(2,3)],4) => [3] => 2
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => [4] => 4
[1,3] => ([(2,3)],4) => [1] => 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [5] => 3
[2,2] => ([(1,3),(2,3)],4) => [2] => 2
[3,1] => ([(0,3),(1,3),(2,3)],4) => [3] => 2
[4] => ([],4) => [] => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 8
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => 6
[1,1,3] => ([(2,3),(2,4),(3,4)],5) => [3] => 2
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [8] => 22
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => [4] => 4
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [5] => 3
[1,4] => ([(3,4)],5) => [1] => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => 3
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [6] => 8
[2,3] => ([(2,4),(3,4)],5) => [2] => 2
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [7] => 6
[3,2] => ([(1,4),(2,4),(3,4)],5) => [3] => 2
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => [4] => 4
[5] => ([],5) => [] => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 8
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 6
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => 22
[1,1,4] => ([(3,4),(3,5),(4,5)],6) => [3] => 2
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => 22
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => [4] => 4
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [5] => 3
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 8
[1,5] => ([(4,5)],6) => [1] => 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [5] => 3
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [6] => 8
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 6
[2,4] => ([(3,5),(4,5)],6) => [2] => 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [7] => 6
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [8] => 22
[3,3] => ([(2,5),(3,5),(4,5)],6) => [3] => 2
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => [4] => 4
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5] => 3
[6] => ([],6) => [] => 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => 8
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7] => 6
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8] => 22
[1,1,5] => ([(4,5),(4,6),(5,6)],7) => [3] => 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8] => 22
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => [4] => 4
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [5] => 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [6] => 8
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => [7] => 6
[1,6] => ([(5,6)],7) => [1] => 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [5] => 3
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [6] => 8
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7] => 6
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8] => 22
[2,5] => ([(4,6),(5,6)],7) => [2] => 2
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [7] => 6
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [8] => 22
[3,4] => ([(3,6),(4,6),(5,6)],7) => [3] => 2
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => [4] => 4
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [5] => 3
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6] => 8
[7] => ([],7) => [] => 1
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Description
The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
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 edge-partition of connected components
Description
Sends a graph to the partition recording the number of edges in its connected components.