Abstract:
Topological indices, sometimes also recognized as a graph-theoretic invariant, maintain
the symmetry of the molecular structures and assign a mathematical language to predict
features such as the radius of gyrations, viscosity, boiling points, etc. The topological
invariant can be considered as a numeric amount which interconnected with a graph which
captures the graph topology and is unchanged under graph automorphism. Nowadays,
topological indices have been developing attention due to their significance in the domain
of computational chemistry. There are certain crucial categories of topological indices with
respect to their specific topological features, like degrees of vertices, distances between
vertices, eccentricities of vertices, connectivity, matching number, etc.
The main focus of this dissertation is to derive extremal graphs with respect to some
eccentricity based indices. We determine the extremal conjugated trees with respect to
eccentric connectivity index and also eccentric adjacency index among all n-vertex con
jugated trees. We focus on the unicyclic graphs with the largest unicyclic graph with
respect to eccentric adjacency index with fixed order and girth. We determine the tree
with the largest eccentric adjacency index among all the trees with a fixed diameter. Next
we derive the extremal trees with the eccentric connectivity and the eccentric adjacency
indices among the trees with a given bipartition size, fixed matching number, fixed in
dependence number and fixed domination number. We obtain the graphs with fixed cut
edges which have the largest eccentric adjacency index and characterized the extremal
graphs. We determine the trees with the smallest and the largest total eccentricity index
among the class of trees with p pendent vertices. Furthermore, we define a class of trees
with a fixed diameter and investigate the trees with the smallest and the largest total
eccentricity index in this class.
