- FindStat

Identifier

Mp00259: Graphs —vertex addition⟶ Graphs
St001723: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 3

([(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) => 2

([(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) => 3

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

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

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

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

([(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) => 2

([(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) => 3

([(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) => 2

([(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) => 2

([(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) => 3

([(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) => 3

([(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) => 2

([(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) => 3

([(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) => 3

([(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) => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 4

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

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

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

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

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

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

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

([(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) => 3

([(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) => 3

([(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) => 3

([(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) => 4

([(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) => 2

([(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) => 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) => 3

([(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) => 3

([(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) => 3

([(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) => 4

([(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) => 3

([(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) => 4

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

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

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

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

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

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

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

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

([(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) => 3

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

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

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

([(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) => 2

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

>>> Load all 208 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 2 + 2\ q$

$F_{4} = 3 + 4\ q + 4\ q^{2}$

$F_{5} = 3 + 8\ q + 12\ q^{2} + 11\ q^{3}$

$F_{6} = 4 + 10\ q + 48\ q^{2} + 60\ q^{3} + 34\ q^{4}$

Description

The differential of a graph.
The external neighbourhood (or boundary) of a set of vertices

$S\subseteq V(G)$ is the set of vertices not in

$S$ which are adjacent to a vertex in

$S$ .
The differential of a set of vertices

$S\subseteq V(G)$ is the difference of the size of the external neighbourhood of

$S$ and the size of

$S$ .
The differential of a graph is the maximal differential of a set of vertices.

Map

vertex addition

Description

Adds a disconnected vertex to a graph.