Identifier
-
Mp00324:
Graphs
—chromatic difference sequence⟶
Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤ
Values
([],1) => [1] => [1] => ([],1) => 0
([],2) => [2] => [2] => ([],2) => 0
([(0,1)],2) => [1,1] => [1,1] => ([(0,1)],2) => 1
([],3) => [3] => [3] => ([],3) => 0
([(1,2)],3) => [2,1] => [1,2] => ([(1,2)],3) => 1
([(0,2),(1,2)],3) => [2,1] => [1,2] => ([(1,2)],3) => 1
([(0,1),(0,2),(1,2)],3) => [1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 2
([],4) => [4] => [4] => ([],4) => 0
([(2,3)],4) => [3,1] => [1,3] => ([(2,3)],4) => 1
([(1,3),(2,3)],4) => [3,1] => [1,3] => ([(2,3)],4) => 1
([(0,3),(1,3),(2,3)],4) => [3,1] => [1,3] => ([(2,3)],4) => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 3
([],5) => [5] => [5] => ([],5) => 0
([(3,4)],5) => [4,1] => [1,4] => ([(3,4)],5) => 1
([(2,4),(3,4)],5) => [4,1] => [1,4] => ([(3,4)],5) => 1
([(1,4),(2,4),(3,4)],5) => [4,1] => [1,4] => ([(3,4)],5) => 1
([(0,4),(1,4),(2,4),(3,4)],5) => [4,1] => [1,4] => ([(3,4)],5) => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 4
([],6) => [6] => [6] => ([],6) => 0
([(4,5)],6) => [5,1] => [1,5] => ([(4,5)],6) => 1
([(3,5),(4,5)],6) => [5,1] => [1,5] => ([(4,5)],6) => 1
([(2,5),(3,5),(4,5)],6) => [5,1] => [1,5] => ([(4,5)],6) => 1
([(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,5] => ([(4,5)],6) => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [5,1] => [1,5] => ([(4,5)],6) => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => [1,1,1,1,1,1] => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 5
([],7) => [7] => [7] => ([],7) => 0
([(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(4,6),(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(3,6),(4,6),(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(2,6),(3,6),(4,6),(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [6,1] => [1,6] => ([(5,6)],7) => 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 6
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Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Map
rotate back to front
Description
The back to front rotation of an integer composition.
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