Values
([(0,1)],2) => ([(0,1)],2) => ([],2) => 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
([(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
([(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
([(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
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
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Description
The normalized isoperimetric number of a graph.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$
where
$$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$
This statistic is $i(G)\cdot\lfloor n/2\rfloor !$. We leave the statistic undefined for the graph without vertices and the graph with a single vertex.
The isoperimetric number, or Cheeger constant, of a graph $G$ is
$$ i(G) = \min\left\{\frac{|\partial A|}{|A|}\ : \ A\subseteq V(G), 0 < |A|\leq |V(G)|/2\right\}, $$
where
$$ \partial A := \{(x, y)\in E(G)\ : \ x\in A, y\in V(G)\setminus A \}. $$
This statistic is $i(G)\cdot\lfloor n/2\rfloor !$. We leave the statistic undefined for the graph without vertices and the graph with a single vertex.
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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