- FindStat

Identifier

Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001308: Graphs ⟶ ℤ

Values

[.,.] => [[],[]] => ([(0,2),(1,2)],3) => ([(0,1)],2) => 0
[.,[.,.]] => [[],[[],[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 2
[[.,.],.] => [[[],[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 2
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 5
[.,[[.,.],.]] => [[],[[[],[]],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 5
[[.,.],[.,.]] => [[[],[]],[[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 3
[[.,[.,.]],.] => [[[],[[],[]]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 5
[[[.,.],.],.] => [[[[],[]],[]],[]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 5
[.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 6
[[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 6
[[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 6
[[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 6
[[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9) => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => 6
[[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 6

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$F_{1} = 1$

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

$F_{3} = q^{3} + 4\ q^{5}$

Description

The number of induced paths on three vertices in a graph.

Map

to complete tree

Description

Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.

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.

Map

to graph

Description

Return the undirected graph obtained from the tree nodes and edges.