Identifier
-
Mp00011:
Binary trees
—to graph⟶
Graphs
St000450: Graphs ⟶ ℤ
Values
[.,.] => ([],1) => 1
[.,[.,.]] => ([(0,1)],2) => 1
[[.,.],.] => ([(0,1)],2) => 1
[.,[.,[.,.]]] => ([(0,2),(1,2)],3) => 1
[.,[[.,.],.]] => ([(0,2),(1,2)],3) => 1
[[.,.],[.,.]] => ([(0,2),(1,2)],3) => 1
[[.,[.,.]],.] => ([(0,2),(1,2)],3) => 1
[[[.,.],.],.] => ([(0,2),(1,2)],3) => 1
[.,[.,[.,[.,.]]]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[.,[[.,.],.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[[.,.],[.,.]]] => ([(0,3),(1,3),(2,3)],4) => 1
[.,[[.,[.,.]],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[[[.,.],.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[.,[.,.]]],.] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[[.,.],.]],.] => ([(0,3),(1,2),(2,3)],4) => 1
[[[.,.],[.,.]],.] => ([(0,3),(1,3),(2,3)],4) => 1
[[[.,[.,.]],.],.] => ([(0,3),(1,2),(2,3)],4) => 1
[[[[.,.],.],.],.] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[.,[.,[.,[.,.]]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[.,[.,[[.,.],.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[.,[[.,[.,.]],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[.,[[[.,.],.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[[.,.],[[.,.],.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[[[.,.],.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[[.,[.,[.,.]]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[[.,[[.,.],.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[.,[[[.,[.,.]],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[[[[.,.],.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,.],[.,[[.,.],.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[.,.],[[.,[.,.]],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,.],[[[.,.],.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[.,.]],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[.,.]],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,.],.],[.,[.,.]]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,.],.],[[.,.],.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[[.,.],.]],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,.],[.,.]],[.,.]] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[.,[.,.]],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[[.,.],.],.],[.,.]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[.,[.,[.,.]]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[.,[[.,.],.]]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[.,[[.,[.,.]],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[.,[[[.,.],.],.]],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,.],[.,[.,.]]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[.,.],[[.,.],.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[.,[.,.]],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[[.,.],.],[.,.]],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[.,[.,[.,.]]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[.,[[.,.],.]],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[[[[.,[.,.]],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[[[[[.,.],.],.],.],.] => ([(0,4),(1,3),(2,3),(2,4)],5) => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[.,[[.,.],[.,.]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[[.,.],[.,[.,.]]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[[.,.],[[.,.],.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[[.,[.,.]],[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[[[.,.],.],[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[[[.,.],[.,.]],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[[.,.],[.,[.,[.,.]]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,.],[.,[[.,.],.]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[.,[[.,.],[[.,[.,.]],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,.],[[[.,.],.],.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,[.,.]],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[.,[.,.]],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[.,[.,[.,.]]],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,[[.,.],.]],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[.,[[[.,[.,.]],.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[[[.,.],.],.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[[.,[[.,.],[.,.]]],.]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 1
[.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[.,[[[.,.],[.,[.,.]]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[.,.],[[.,.],.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[.,[.,.]],[.,.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[[.,.],.],[.,.]],.]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 1
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Description
The number of edges minus the number of vertices plus 2 of a graph.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
When G is connected and planar, this is also the number of its faces.
When $G=(V,E)$ is a connected graph, this is its $k$-monochromatic index for $k>2$: for $2\leq k\leq |V|$, the $k$-monochromatic index of $G$ is the maximum number of edge colors allowed such that for each set $S$ of $k$ vertices, there exists a monochromatic tree in $G$ which contains all vertices from $S$. It is shown in [1] that for $k>2$, this is given by this statistic.
Map
to graph
Description
Return the undirected graph obtained from the tree nodes and edges, with leaves being ignored.
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