Identifier
Values
[1] => [1] => ([],1) => ([],1) => 1
[1,2] => [1,2] => ([],2) => ([],1) => 1
[2,1] => [1,2] => ([],2) => ([],1) => 1
[1,2,3] => [1,2,3] => ([],3) => ([],1) => 1
[1,3,2] => [1,2,3] => ([],3) => ([],1) => 1
[2,1,3] => [1,2,3] => ([],3) => ([],1) => 1
[2,3,1] => [1,2,3] => ([],3) => ([],1) => 1
[1,2,3,4] => [1,2,3,4] => ([],4) => ([],1) => 1
[1,2,4,3] => [1,2,3,4] => ([],4) => ([],1) => 1
[1,3,2,4] => [1,2,3,4] => ([],4) => ([],1) => 1
[1,3,4,2] => [1,2,3,4] => ([],4) => ([],1) => 1
[2,1,3,4] => [1,2,3,4] => ([],4) => ([],1) => 1
[2,1,4,3] => [1,2,3,4] => ([],4) => ([],1) => 1
[2,3,1,4] => [1,2,3,4] => ([],4) => ([],1) => 1
[2,3,4,1] => [1,2,3,4] => ([],4) => ([],1) => 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,3,5,4] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,4,3,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,4,5,3] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,3,2,4,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,3,2,5,4] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,3,4,2,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,3,4,5,2] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,1,3,4,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,1,3,5,4] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,1,4,3,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,1,4,5,3] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,3,1,4,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,3,1,5,4] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,3,4,1,5] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[2,3,4,5,1] => [1,2,3,4,5] => ([],5) => ([],1) => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,3,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,4,3,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,4,3,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,4,5,3,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,1,4,5,6,3] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,1,4,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,1,4,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,1,5,4,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,1,5,6,4] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,4,1,5,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,4,1,6,5] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,4,5,1,6] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([],6) => ([],1) => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,3,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,3,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,3,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,3,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,3,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,5,3,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,5,3,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,2,4,5,6,7,3] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,2,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,2,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,2,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,2,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,2,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,5,2,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,5,2,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[1,3,4,5,6,7,2] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
>>> Load all 127 entries. <<<
[2,1,3,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,3,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,3,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,3,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,3,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,3,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,5,3,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,5,3,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,5,6,3,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,1,4,5,6,7,3] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,4,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,4,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,4,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,4,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,5,4,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,5,4,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,5,6,4,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,1,5,6,7,4] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,1,5,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,1,5,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,1,6,5,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,1,6,7,5] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,5,1,6,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,5,1,7,6] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,5,6,1,7] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
[2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([],7) => ([],1) => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
de-duplicate
Description
The de-duplicate of a graph.
Let $G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods $\{N_v | v \in V\}$ of $G$, and has an edge $(N_a, N_b)$ between two vertices if and only if $(a, b)$ is an edge of $G$. This is well-defined, because if $N_a = N_c$ and $N_b = N_d$, then $(a, b)\in E$ if and only if $(c, d)\in E$.
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.