- FindStat

Identifier

Mp00259: Graphs —vertex addition⟶ Graphs
Mp00318: Graphs —dual on components⟶ Graphs
St001496: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 1

([(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) => 1

([(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) => 1

([(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) => 1

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

([(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) => 1

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

([(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) => 1

([(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) => 1

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

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

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

([(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) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of graphs with the same Laplacian spectrum as the given graph.

Map

vertex addition

Description

Adds a disconnected vertex to a graph.

Map

dual on components

Description

The dual of the 3-connected planar components of a graph.
Replace each connected component which is 3-connected and planar with its planar dual.