Extremal values of vertex--degree--based topological indices over
fluoranthene-type benzenoid systems with equal number of edges
Abstract
Let $G$ be a graph with $n$ vertices. A vertex–degree–based
topological index is defined from a set of real numbers
$\{\psi_{ij}\}$ as
$$TI(G)=\sum_{1\leq
i\leq j\leq
n-1}m_{ij}\psi_{ij},$$ where $m_{ij}$ is
the number of edges of $G$ connecting a vertex of degree $i$ with a
vertex of degree $j$. Many of the well–known topological indices are
particular cases of this expression, including all
Randi\’{c}-type connectivity indices. In this work we
determine extremal values for $TI$ over the set of fluoranthene–type
benzenoid systems with a fixed number of edges. The main idea consists
in constructing fluoranthene–type benzenoid systems with maximal number
of inlets in $\Gamma_{m}$ which have simultaneously
minimal number of hexagons, where $\Gamma_{m}$ is
the set of fluoranthene–type benzenoid systems with exactly
$m(m\geq19)$ edges.