- FindStat

Identifier

Mp00274: Graphs —block-cut tree⟶ Graphs
St000466: Graphs ⟶ ℤ

Values

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

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

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

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

([(0,3),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 15

([(0,3),(1,2),(2,3)],4) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

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

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

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

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,4),(3,4)],5) => 28

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 73

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,3),(2,3)],4) => 15

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

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

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

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

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 146

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

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

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

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

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

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

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

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

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

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

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

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 45

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => 106

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,4),(2,4),(3,4)],5) => 28

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 114

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 15

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 73

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 15

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

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

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

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 73

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 15

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

([(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) => 114

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

([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 146

([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 146

([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 73

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

([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 44

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

([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 15

([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,3),(2,3)],4) => 15

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

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

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

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

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

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

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

([(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) => 44

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

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

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

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

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

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

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

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

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

([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([],1) => 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) => 44

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

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

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

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

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

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

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

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

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

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

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

([(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) => 6

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

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

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

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

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

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

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

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$F_{1} = 1$

Description

The Gutman (or modified Schultz) index of a connected graph.
This is

$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$
where

$d(u)$ is the degree of vertex

$u$ and

$d(u,v)$ is the distance between vertices

$u$ and

$v$ .
For trees on

$n$ vertices, the modified Schultz index is related to the Wiener index via

$S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].

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.