Processing math: 100%

Identifier
Values
([],1) => [1] => [1] => ([],1) => 1
([],2) => [2] => [1,1] => ([(0,1)],2) => 2
([],3) => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 3
([],4) => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 4
([],5) => [5] => [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) => 5
([],6) => [6] => [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) => 6
([],7) => [7] => [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) => 7
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Description
The pebbling number of a connected 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.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let G be a simple graph with chromatic number κ. Let αm be the maximum number of vertices in a m-colorable subgraph of G. Set δm=αmαm1. The sequence δ1,δ2,δκ is the chromatic difference sequence of G.
All entries of the chromatic difference sequence are positive: αm>αm1 for m<κ, because we can assign any uncolored vertex of a partial coloring with m1 colors the color m. Therefore, the chromatic difference sequence is a composition of the number of vertices of G into κ parts.
Map
complement
Description
The complement of a composition.
The complement of a composition I is defined as follows:
If I is the empty composition, then the complement is also the empty composition. Otherwise, let S be the descent set corresponding to I=(i1,,ik), that is, the subset
{i1,i1+i2,,i1+i2++ik1}
of {1,2,,|I|1}. Then, the complement of I is the composition of the same size as I, whose descent set is {1,2,,|I|1}S.
The complement of a composition I coincides with the reversal (Mp00038reverse) of the composition conjugate (Mp00041conjugate) to I.