Values
([],1) => ([],1) => ([],1) => 0
([],2) => ([],1) => ([],1) => 0
([(0,1)],2) => ([(0,1)],2) => ([],2) => 0
([],3) => ([],1) => ([],1) => 0
([(1,2)],3) => ([(0,1)],2) => ([],2) => 0
([(0,2),(1,2)],3) => ([(0,1)],2) => ([],2) => 0
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([],4) => ([],1) => ([],1) => 0
([(2,3)],4) => ([(0,1)],2) => ([],2) => 0
([(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 0
([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 0
([(0,3),(1,2)],4) => ([(0,1)],2) => ([],2) => 0
([(0,3),(1,2),(2,3)],4) => ([(0,1)],2) => ([],2) => 0
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([],2) => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(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) => 0
([],5) => ([],1) => ([],1) => 0
([(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,3)],5) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(0,1),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1)],2) => ([],2) => 0
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(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) => 0
([(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) => 0
([(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) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(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) => 0
([(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) => 0
([(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) => 0
([],6) => ([],1) => ([],1) => 0
([(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(2,5),(3,4)],6) => ([(0,1)],2) => ([],2) => 0
([(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,2),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,1)],2) => ([],2) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(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) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(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) => 0
([(0,5),(1,4),(2,3)],6) => ([(0,1)],2) => ([],2) => 0
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,1)],2) => ([],2) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
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Description
The determinant of the adjacency matrix of a graph.
For a labelled graph $G$ with vertices $\{1,\ldots,n\}$, the adjacency matrix is the matrix $(a_{ij})$ with $a_{ij} = 1$ if the vertices $i$ and $j$ are joined by an edge in $G$.
Since the determinant is invariant under simultaneous row and column permutations, the determinant of the adjacency is well-defined for an unlabelled graph.
According to [2], Equation 8, this determinant can be computed as follows: let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that equal $K_2$. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs $H$ of $G$ that have as connected components only $K_2$'s and cycles.
For a labelled graph $G$ with vertices $\{1,\ldots,n\}$, the adjacency matrix is the matrix $(a_{ij})$ with $a_{ij} = 1$ if the vertices $i$ and $j$ are joined by an edge in $G$.
Since the determinant is invariant under simultaneous row and column permutations, the determinant of the adjacency is well-defined for an unlabelled graph.
According to [2], Equation 8, this determinant can be computed as follows: let $s(G)$ be the number of connected components of $G$ that are cycles and $r(G)$ the number of connected components that equal $K_2$. Then $$\det(A) = \sum_{H} (-1)^{r(H)} 2^{s(H)}$$ where the sum is over all spanning subgraphs $H$ of $G$ that have as connected components only $K_2$'s and cycles.
Map
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
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
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