Identifier
Values
[1,1] => [1,1] => ([(0,1)],2) => ([],1) => 1
[1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 1
[1,2] => [1,2] => ([(1,2)],3) => ([],1) => 1
[2,1] => [2,1] => ([(0,2),(1,2)],3) => ([(0,1)],2) => 1
[1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 1
[1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2
[1,3] => [1,3] => ([(2,3)],4) => ([],1) => 1
[2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[2,2] => [2,2] => ([(1,3),(2,3)],4) => ([(0,1)],2) => 1
[3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 1
[1,1,1,2] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,4] => [1,4] => ([(3,4)],5) => ([],1) => 1
[2,1,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 2
[2,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,3] => [2,3] => ([(2,4),(3,4)],5) => ([(0,1)],2) => 1
[3,1,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,2] => [3,2] => ([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 1
[4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[1,1,1,3] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,2,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 2
[1,1,4] => [1,1,4] => ([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
[1,2,3] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,5] => [1,5] => ([(4,5)],6) => ([],1) => 1
[2,1,3] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2
[2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[2,4] => [2,4] => ([(3,5),(4,5)],6) => ([(0,1)],2) => 1
[3,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 1
[4,1,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(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) => 2
[4,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(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
[1,1,1,4] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 2
[1,1,5] => [1,1,5] => ([(4,5),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => 1
[1,2,4] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2
[1,3,3] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2
[1,6] => [1,6] => ([(5,6)],7) => ([],1) => 1
[2,1,4] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2
[2,2,3] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2
[2,5] => [2,5] => ([(4,6),(5,6)],7) => ([(0,1)],2) => 1
[3,2,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 2
[3,4] => [3,4] => ([(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(1,2)],3) => 1
[4,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(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) => 2
[4,3] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 1
[5,2] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
[6,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(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
search for individual values
searching the database for the individual values of this statistic
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number St001337The upper domination number of a graph. has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
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.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00314Foata bijection.
Map
line graph
Description
The line graph of a graph.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.