Identifier
Mp00154:
Graphs
—core⟶
Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Images
([],1) => ([],1) => [1] => [1]
([],2) => ([],1) => [1] => [1]
([(0,1)],2) => ([(0,1)],2) => [2] => [1,1]
([],3) => ([],1) => [1] => [1]
([(1,2)],3) => ([(0,1)],2) => [2] => [1,1]
([(0,2),(1,2)],3) => ([(0,1)],2) => [2] => [1,1]
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([],4) => ([],1) => [1] => [1]
([(2,3)],4) => ([(0,1)],2) => [2] => [1,1]
([(1,3),(2,3)],4) => ([(0,1)],2) => [2] => [1,1]
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => [2] => [1,1]
([(0,3),(1,2)],4) => ([(0,1)],2) => [2] => [1,1]
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => [2] => [1,1]
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => [2] => [1,1]
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1]
([],5) => ([],1) => [1] => [1]
([(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(2,4),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(2,3)],5) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => [2] => [1,1]
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1]
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1]
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1]
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1]
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [4] => [1,1,1,1]
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [5] => [1,1,1,1,1]
([],6) => ([],1) => [1] => [1]
([(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(2,5),(3,4)],6) => ([(0,1)],2) => [2] => [1,1]
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1]
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => [2] => [1,1]
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [3] => [1,1,1]
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [5] => [1,1,1,1,1]
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core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
Map
to partition of connected components
Description
Return the partition of the sizes of the connected components of the graph.
Map
conjugate
Description
Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.
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