Identifier
Values
([],2) => ([],2) => ([(0,1)],2) => 2
([],3) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,2)],3) => ([],2) => ([(0,1)],2) => 2
([],4) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(2,3)],4) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,3),(1,2)],4) => ([],2) => ([(0,1)],2) => 2
([(0,3),(1,2),(2,3)],4) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,2),(1,3),(2,3)],4) => ([],2) => ([(0,1)],2) => 2
([],5) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120
([(3,4)],5) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(1,4),(2,3)],5) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(0,1),(2,4),(3,4)],5) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,3),(2,4),(3,4)],5) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,3),(1,4),(2,3),(2,4)],5) => ([],2) => ([(0,1)],2) => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],2) => ([(0,1)],2) => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,1),(2,3),(2,4),(3,4)],5) => ([],2) => ([(0,1)],2) => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],2) => ([(0,1)],2) => 2
([(4,5)],6) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120
([(2,5),(3,4)],6) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(1,2),(3,5),(4,5)],6) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(3,4),(3,5),(4,5)],6) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,4),(2,5),(3,4),(3,5)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,5),(1,4),(2,3)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,2),(3,4),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([],2) => ([(0,1)],2) => 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([],2) => ([(0,1)],2) => 2
([(3,6),(4,5)],7) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120
([(4,5),(4,6),(5,6)],7) => ([],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 120
([(3,5),(3,6),(4,5),(4,6)],7) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([],2) => ([(0,1)],2) => 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([],2) => ([(0,1)],2) => 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 24
([(1,6),(2,5),(3,4)],7) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(0,3),(1,2),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(2,3),(4,5),(4,6),(5,6)],7) => ([],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 24
([(1,2),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 12
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 6
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) => ([(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 48
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 4
>>> Load all 284 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
For a connected graph $G$ the $k$-th Elser number is
$$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
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.
Map
block-cut tree
Description
Sends a graph to its block-cut tree.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
The block-cut tree has a vertex for each block and for each cut-vertex of the given graph, and there is an edge for each pair of block and cut-vertex that belongs to that block. A block is a maximal biconnected (or 2-vertex connected) subgraph. A cut-vertex is a vertex whose removal increases the number of connected components.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!