GutmanIndex

The sum over the pairs of vertices of the product of their degrees times their distance.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000466via LabeledGraph✓↗

GutmanIndex(graph)The sum over the pairs of vertices of the product of their degrees times their distance.

Typed signatures

Type	Package	Applies
(labeled_graph) -> number	combinatorics	always

Domain: Graph statistics

Details

• Defined over LabeledGraph as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a LabeledGraph and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• FindStat's St000466, the modified Schultz index, of a connected graph; a pair with no path adds nothing. The graph with no vertices gives 0.

Definition

Epsil, as the implementation runs it.

on LabeledGraph, over _x

0 if _x[1] == 0 else Apply((n: integer) => Apply((edges: list<list<integer>>) => Apply(
                                                                                        (degrees: list<integer>) => Apply(
                                                                                                                           (adjacency: list<list<integer>>) => Apply(
                                                                                                                                                                      (distances: list<list<integer>>) => Max(Sum(Map(Function(Max(Sum(Map(Function(degrees[u] * degrees[w] *
                                                                                                                                                                                                                                                    distances[u][w] if distances[u][w] < n else 0, w), Range(1, n))), 0), u), Range(1, n))), 0) /
                                                                                                                                                                                                          2,
                                                                                                                                                                      Apply((matrix: list<list<integer>>) => Apply(
                                                                                                                                                                                                                    (a_init: list<list<integer>>) => Fold((a: list<list<integer>>, k) => Apply(
                                                                                                                                                                                                                                                                                                (a_next: list<list<integer>>) => a_next,
                                                                                                                                                                                                                                                                                                Map(Function(Map(Function(Min(a[i][j], a[i][k] + a[k][j]), j), Range(1, n)), i), Range(1, n)),
                                                                                                                                                                                                                                                                                              ), a_init, Range(1, n)),
                                                                                                                                                                                                                    Map(Function(Map(Function(0 if i == j else 1 if matrix[i][j] == 1 else 2n, j), Range(1, n)), i), Range(1, n)),
                                                                                                                                                                                                                  ), adjacency),
                                                                                                                                                                    ),
                                                                                                                           Map(
                                                                                                                                Function(Map(Function(Max(Sum(Map(Function((1 if e[1] == i else 0) *
                                                                                                                                                                           (1 if e[2] == j else 0) +
                                                                                                                                                                           (1 if e[1] == j else 0) *
                                                                                                                                                                           (1 if e[2] == i else 0), e), edges)), 0), j), Range(1, n)), i),
                                                                                                                                Range(1, n),
                                                                                                                              ),
                                                                                                                         ),
                                                                                        Map(
                                                                                             Function(Max(Sum(Map(Function((1 if e[1] == v else 0) +
                                                                                                                           (1 if e[2] == v else 0), e), edges)), 0), v),
                                                                                             Range(1, n),
                                                                                           ),
                                                                                      ), _x[2]), _x[1])

Examples

Basic #

the graph on 4 vertices with the edges 12, 23, 24 and 34

live ↗

In$\mathrm{GutmanIndex}(\operatorname{LabeledGraph}((4,\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack)))$Out$27$

Properties #

FindStat's value for ([],1)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((1,\bigl\lbrack\bigr\rbrack)), \text{GutmanIndex})$Out$0$

Known value · FindStat St000466

FindStat's value for ([(0,1)],2)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((2,\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack)), \text{GutmanIndex})$Out$1$

Known value · FindStat St000466

FindStat's value for ([(0,2),(1,2)],3)

live ↗

In$\mathrm{CombinatorialStat}(\operatorname{LabeledGraph}((3,\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 3\bigr\rbrack\bigr\rbrack)), \text{GutmanIndex})$Out$6$

Known value · FindStat St000466

About

Examples	4
Crosswalk	FindStat