- FindStat

Identifier

Mp00193: Lattices —to poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000718: Graphs ⟶ ℤ

Values

([],1) => ([],1) => ([],1) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => ([(0,1)],2) => 2
([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => ([(0,1)],2) => 2
([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => ([(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) => 2
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => ([(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) => 2
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],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) => 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,1)],2) => 2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => ([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => ([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => ([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 4
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => ([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 5

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$F_{1} = 1$

$F_{2} = q^{2}$

$F_{3} = q^{2}$

Description

The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]

Map

to graph

Description

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

Map

to poset

Description

Return the poset corresponding to the lattice.

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.