Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St000772
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1 => [1] => ([],1)
 => 1
[2,1] => 0 => [1] => ([],1)
 => 1
[1,3,2] => 10 => [1,1] => ([(0,1)],2)
 => 1
[2,1,3] => 01 => [1,1] => ([(0,1)],2)
 => 1
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,4,6,5] => 10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,2,5,4,6] => 10101 => [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
[1,3,4,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,3,4,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,2,4,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,2,5,3,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,6,5] => 10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,4,3,5,2,6] => 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
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].