Identifier
Values
([],1) => [1] => ([],1) => 0
([],2) => [2] => ([],2) => 0
([(0,1)],2) => [1,1] => ([(0,1)],2) => 2
([],3) => [3] => ([],3) => 0
([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 4
([],4) => [4] => ([],4) => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 6
([],5) => [5] => ([],5) => 0
([(3,4)],5) => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
([(2,4),(3,4)],5) => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
([(1,4),(2,4),(3,4)],5) => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
([(0,4),(1,4),(2,4),(3,4)],5) => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
([(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 6
([(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] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8
([],6) => [6] => ([],6) => 0
([(2,5),(3,4)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(2,5),(3,4),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(1,2),(3,5),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(1,5),(2,5),(3,4),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,1),(2,5),(3,5),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,5),(2,4),(3,4)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(1,5),(2,4),(3,4),(3,5)],6) => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => 4
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 6
([(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] => ([(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) => 10
([],7) => [7] => ([],7) => 0
([(1,6),(2,5),(3,4)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,2),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,3),(1,2),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(2,6),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,6),(2,5),(3,4),(3,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,5),(2,4),(3,4),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(0,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,1),(2,5),(3,4),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7) => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => 4
([(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] => ([(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) => 12
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Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
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.
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.