## Bounds on distance-based topological indices in graphs.

##### Abstract

This thesis details the results of investigations into bounds on some distance-based
topological indices.
The thesis consists of six chapters. In the first chapter we define the standard
graph theory concepts, and introduce the distance-based graph invariants called
topological indices. We give some background to these mathematical models, and
show their applications, which are largely in chemistry and pharmacology. To complete
the chapter we present some known results which will be relevant to the work.
Chapter 2 focuses on the topological index called the eccentric connectivity index.
We obtain an exact lower bound on this index, in terms of order, and show that this
bound is sharp. An asymptotically sharp upper bound is also derived. In addition,
for trees of given order, when the diameter is also prescribed, tight upper and lower
bounds are provided.
Our investigation into the eccentric connectivity index continues in Chapter 3.
We generalize a result on trees from the previous chapter, proving that the known
tight lower bound on the index for a tree in terms of order and diameter, is also
valid for a graph of given order and diameter.
In Chapter 4, we turn to bounds on the eccentric connectivity index in terms of
order and minimum degree. We first consider graphs with constant degree (regular
graphs). Došlić, Saheli & Vukičević, and Ilić posed the problem of determining
extremal graphs with respect to our index, for regular (and more specifically,
cubic) graphs. In addressing this open problem, we find upper and lower bounds
for the index. We also provide an extremal graph for the upper bound. Thereafter,
the chapter continues with a consideration of minimum degree. For given order and
minimum degree, an asymptotically sharp upper bound on the index is derived.
In Chapter 5, we turn our focus to the well-studied Wiener index. For trees
of given order, we determine a sharp upper bound on this index, in terms of the
eccentric connectivity index. With the use of spanning trees, this bound is then
generalized to graphs.
Yet another distance-based topological index, the degree distance, is considered
in Chapter 6. We find an asymptotically sharp upper bound on this index, for a
graph of given order. This proof definitively settles a conjecture posed by Tomescu
in 1999.