### G-2018-69

# Minimum eccentric connectivity index for graphs with fixed order and fixed number of pending vertices

## Gauvain Devillez, Alain Hertz, Hadrien Mélot, and Pierre Hauweele

The eccentric connectivity
index of a connected graph `\(G\)`

is the sum over all vertices `\(v\)`

of the product `\(d_G(v)e_G(v)\)`

, where `\(d_G(v)\)`

is the degree of `\(v\)`

in `\(G\)`

and `\(e_G(v)\)`

is the maximum distance
between `\(v\)`

and any other vertex of `\(G\)`

. This index is helpful for the prediction of biological activities
of diverse nature, a molecule being modeled as a graph where
atoms are represented by vertices and
chemical bonds by edges.
We characterize those graphs which
have the smallest eccentric connectivity index among all connected graphs of a given order `\(n\)`

. Also, given two integers `\(n\)`

and `\(p\)`

with `\(p\leq n-1\)`

, we characterize those graphs which have the smallest
eccentric connectivity index among all connected graphs of order `\(n\)`

with `\(p\)`

pending vertices.

Published **September 2018**
,
11 pages