- FindStat

Identifier

Mp00259: Graphs —vertex addition⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000447: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => ([],6) => ([],1) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([],7) => ([],1) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

$F_{4} = 10 + q$

$F_{5} = 27 + 5\ q + 2\ q^{2}$

Description

The number of pairs of vertices of a graph with distance 3.
This is the coefficient of the cubic term of the Wiener polynomial, also called Wiener polarity index.

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

vertex addition

Description

Adds a disconnected vertex to a graph.