- FindStat

Identifier

Mp00274: Graphs —block-cut tree⟶ Graphs
St000455: Graphs ⟶ ℤ

Values

([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 0

([(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 0

([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,3),(1,2),(2,3)],4) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(1,2)],3) => 0

([(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 0

([(1,4),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 0

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0

([(1,4),(2,3),(3,4)],5) => ([(1,5),(2,4),(3,4),(3,5)],6) => 1

([(0,1),(2,4),(3,4)],5) => ([(1,3),(2,3)],4) => 0

([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,3),(2,3)],4) => 0

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 0

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 0

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 0

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 0

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 0

([(3,5),(4,5)],6) => ([(3,5),(4,5)],6) => 0

([(2,5),(3,5),(4,5)],6) => ([(2,5),(3,5),(4,5)],6) => 0

([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0

([(2,5),(3,4),(4,5)],6) => ([(2,6),(3,5),(4,5),(4,6)],7) => 1

([(1,2),(3,5),(4,5)],6) => ([(2,4),(3,4)],5) => 0

([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,4),(3,4)],5) => 0

([(2,5),(3,4),(3,5),(4,5)],6) => ([(2,4),(3,4)],5) => 0

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,4),(3,4)],5) => 0

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,4),(2,4),(3,4)],5) => 0

([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,5),(1,5),(2,4),(3,4)],6) => 0

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 1

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 0

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 1

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 1

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 1

([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(1,3),(2,3)],4) => 0

([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 0

([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 0

([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 0

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,2)],3) => 0

([(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) => 1

([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 1

([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(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) => 1

([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(1,2)],3) => 0

([(4,6),(5,6)],7) => ([(4,6),(5,6)],7) => 0

([(3,6),(4,6),(5,6)],7) => ([(3,6),(4,6),(5,6)],7) => 0

([(2,6),(3,6),(4,6),(5,6)],7) => ([(2,6),(3,6),(4,6),(5,6)],7) => 0

([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0

([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0

([(2,3),(4,6),(5,6)],7) => ([(3,5),(4,5)],6) => 0

([(1,2),(3,6),(4,6),(5,6)],7) => ([(2,5),(3,5),(4,5)],6) => 0

([(3,6),(4,5),(4,6),(5,6)],7) => ([(3,5),(4,5)],6) => 0

([(0,1),(2,6),(3,6),(4,6),(5,6)],7) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0

([(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,5),(3,5),(4,5)],6) => 0

([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,5),(2,5),(3,5),(4,5)],6) => 0

([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0

([(1,6),(2,6),(3,5),(4,5)],7) => ([(1,6),(2,6),(3,5),(4,5)],7) => 0

([(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => 0

([(2,6),(3,5),(4,5),(4,6),(5,6)],7) => ([(2,6),(3,5),(4,5),(4,6)],7) => 1

([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => ([(1,4),(2,4),(3,4)],5) => 0

([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(2,4),(3,4)],5) => 0

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search for individual values

searching the database for the individual values of this statistic

Description

The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.

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.