Extremal Instances of Polyphenylene Dendrimers for the Merrifield-Simmons Index

Abstract

Computing topological indices for molecular
graphs is a key element in computational chemistry,
particularly when analyzing the structural and functional
properties of chemical compounds. Among these
indices, the Merrifield-Simmons (M-S) index, defined
as the total number of independent vertex sets in a
molecular graph, provides valuable information on the
compartmentalization, stability, and connectivity of a
molecular graph. These properties are essential when
dendrimer structures are used in nanotechnology, drug
delivery systems, and advanced material design.
Dendrimers, especially polyphenylene dendrimers
(PPDs), are highly branched macromolecules known
for their robustness, shape persistence, and ability
to encapsulate and release therapeutic agents in a
controlled manner. This paper recognizes the patterns
for extreme topologies associated with the M-S index
on dendrimer graph molecules, analyzing its structural
features that maximize or minimize this topological
invariant.
This study shows how graph theory and recurrence
relations can be used to design efficient counting
algorithms, benefiting both the pattern recognition area
and practical applications in molecular design, drug
delivery, and nanomaterials engineering.