### Abstract:

In mathematical chemistry, a topological index TI is a numerical value which describes
the topology and the properties of a molecular structure. In QSAR/QSPR
studies, topological indices are utilized to predict the biological activity of chemical
compounds. Topological indices are divided as degree based, distance based and computational
based and depending on these types, many topological indices have been
worked out till now.
Wiener index is the oldest distance based graph invariant which is defined as sum of
the distance between all unordered pair of vertices of the graph G. Another important
distance based topological index is the Harary index which is defined as sum of
inverse of the distances between all unordered pair of vertices of G. Automorphism
can be used to calculate the Harary index. The first chapter will consist of basic
definitions and notations which are used throughout this dissertation. In the second
chapter, we compute the exact formula for the Harary index of the carbon nanotube
TUC4C8[p, q].
In third chapter, we will use the method of edge cut to compute the Wiener index of
linear (Ln) and zig-zag (Zn) polyomino chain of 4k − cycles. Also we will introduce
another polyomino chain of 4k-cycle Zmn and calculate its Wiener index for m = 3
and m = 4.
In the forth chapter of this dissertation, we will compute an explicit formulae for the
Sum-connectivity index and Harmonic index of a nanotube TUC5C8[p, q]. We will
also discuss the effect of sub-divisions of the edges of nanotube TUC5C8[p, q] on some
degree based topological indices