Identifier
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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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.
This is the coefficient of the cubic term of the Wiener polynomial, also called Wiener polarity index.
Map
vertex addition
Description
Adds a disconnected vertex to a graph.
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.
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.
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