- FindStat

Identifier

Mp00074: Posets —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000771: Graphs ⟶ ℤ

Values

([],1) => ([],1) => ([],1) => 1

([],2) => ([],2) => ([],1) => 1

([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 1

([],3) => ([],3) => ([],1) => 1

([(0,1),(0,2)],3) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 1

([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 1

([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 1

([],4) => ([],4) => ([],1) => 1

([(0,1),(0,2),(0,3)],4) => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1

([(0,2),(0,3),(3,1)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 1

([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1

([(0,3),(1,3),(3,2)],4) => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1

([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => 1

([(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 1

([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 1

([],5) => ([],5) => ([],1) => 1

([(0,1),(0,2),(0,3),(0,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1

([(0,2),(0,3),(0,4),(4,1)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1

([(0,3),(0,4),(4,1),(4,2)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(3,2),(4,1)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1

([(0,4),(4,1),(4,2),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,4),(4,2),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,4),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,4),(2,3),(4,2)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,3),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,4),(2,3),(2,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,2),(1,4),(2,3)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1

([(0,4),(1,2),(1,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,4),(3,1),(4,3)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,4),(1,2),(1,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(1,2),(1,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => 1

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2

([(0,3),(1,2),(1,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(3,4),(4,1),(4,2)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,2),(2,4),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(3,2),(4,1),(4,3)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(1,2),(2,3),(2,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([],6) => ([],6) => ([],1) => 1

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6) => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 2

([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,1)],2) => 1

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2

([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1

([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 1

([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,4),(0,5),(5,1),(5,2),(5,3)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 1

([(0,4),(0,5),(4,3),(5,1),(5,2)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6) => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6) => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => 1

([(0,5),(5,1),(5,2),(5,3),(5,4)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,5),(1,5),(5,2),(5,3),(5,4)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,5),(1,5),(2,5),(5,3),(5,4)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => 1

([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 1

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Description

The largest multiplicity of a 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 $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.

Map

to graph

Description

Returns the Hasse diagram of the poset as an undirected graph.

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.