Abstract:
In this paper, we propose a geometrical interpretation of bond
additive indices, especially Szeged type indices.
Building on this,
we introduce a new class of bond additive indices, namely the
Szeged–Sombor index, and study its properties.
We determine the
Szeged–Sombor index for elementary graphs and determine its
relationship with other topological indices.
Additionally, we derive
an explicit expression for the Szeged–Sombor index of the Cartesian
product of graphs.
We then establish both upper and lower bounds for the Szeged–Sombor index of
trees and bipartite graphs in terms
of their order
$n$,
and characterize the graphs that attain these bounds.
Additionally, we provide an upper bound
for the Szeged–Sombor index of unicyclic graphs with a fixed order, and
identify the extremal graphs.
We also present
an upper bound on the Szeged–Sombor index of a graph
$G$
in terms of
$n$,
$m$
and
$Sz(G)$,
and characterize the
extremal graphs.
We conclude our study by discussing the chemical significance
of Szeged–Sombor index by analyzing
its values on octane isomers and benzenoid hydrocarbons.
We demonstrate that
the newly proposed Szeged–Sombor index
shows a significantly higher correlation with the chemical properties of the
compounds compared to some other
distance-based topological indices.
Finally, we propose some open problems for
future research.
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