Identifier
Values
[[],[[]]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 2
[[[]],[]] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[]]]] => ([(0,3),(1,2),(2,3)],4) => ([(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
[[],[[]],[]] => ([(0,4),(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
[[],[[[]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[]],[],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 2
[[[]],[[]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[],[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[]]],[]] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[],[[]]]] => ([(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
[[[[],[]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[[]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[[[],[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[[[[]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[]],[[],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[]],[[[]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[],[]],[[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[[]]],[[]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[],[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[[[]]]],[]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[],[],[[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[],[[]],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[],[[],[]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[],[[[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[[]],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[],[]],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[[]]],[]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[],[[]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[[]],[]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[[[],[]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[[[[]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[],[[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(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) => 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) => 2
[[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[],[[[[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 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) => 2
[[],[[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 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) => 2
[[],[[],[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(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) => 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) => 2
[[],[[[],[]]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[[[[]]]],[]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[],[[],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[],[[],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 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) => 2
[[],[[[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 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) => 2
[[],[[[],[],[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[],[[[[],[]]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[]],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[[]],[],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 2
[[[]],[],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 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) => 2
[[[]],[[]],[],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 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) => 2
[[[]],[[],[],[]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
[[[]],[[[],[]]]] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 2
[[[],[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => ([(0,3),(1,2),(2,3)],4) => 2
[[[[]]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 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) => 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) => 2
[[[],[]],[[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => 2
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search for individual values
searching the database for the individual values of this statistic
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
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.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges.
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