Abstract:
Many topological indices which are being used by researchers in the quantitative
structure-property relationship (QSPR) and quantitative structure-activity relationship
(QSAR) studies to predict the physico-chemical properties of molecules, are based
on vertex degrees of the corresponding molecular graphs. When a new topological
index is introduced in chemical graph theory, one of the important questions that need
to be answered is for which members of a certain class of n-vertex graphs this index
assumes minimal and maximal values? On the other hand, there are many well known
graph families and vertex-degree-based topological indices in the literature for which
this question remains open. The main purpose of current study is to address this open
question for some well known families of graphs.
Firstly, the collection of all k-polygonal chains (for k = 3, 4, 5) with fixed number of
k-polygons is considered and the extremal elements from this family are characterized
with respect to several well known bond incident degree (BID) indices (BID indices
form a subclass of the class of all vertex-degree-based topological indices). From the
derived results, many already reported results are obtained as corollaries. Furthermore,
the extremal 4-polygonal (polyomino) chains for some renowned vertex-degree-based
topological indices (other than BID indices) are also determined.
Next, the problem of characterizing the extremal cacti over the certain classes of cacti
(tree-like polyphenylene systems, spiro hexagonal systems and general cacti) with some
fixed parameters is addressed for various well known vertex-degree-based topological
indices.
Finally, some mathematical properties of the atom-bond connectivity index and
augmented Zagreb index are explored
