Molecular descriptors called topological indices are graph invariants that playrna signi_cant role in chemistry, materials science, pharmaceutical sciences andrnengineering, since they can be correlated with a large number of physico-rnchemical properties of molecules. Topological indices are used in the processrnof correlating the chemical structures with various characteristics such asrnboiling points and molar heats of formation. Graph theory is used to char-rnacterize these chemical structures.rnBinary and m-ary trees have extensive applications in chemistry and com-rnputer science, since these trees can represent chemical structures and var-rnious useful networks. In this thesis, we present exact values of all im-rnportant distance-based indices for complete m-ary trees and we introducernthe general eccentric connectivity index of a graph G which is de_ned as,rnECI_(G) =rnPrnv2V (G) eccG(v)d_rnG(v) for _ 2 R, where V (G) is the vertex setrnof G, eccG(v) is the eccentricity of v and dG(v) is the degree of v in G.rnThe thesis consists of four chapters. In the _rst chapter we de_ne the stan-rndard graph theory concepts, and introduce the distance-based graph invari-rnants called topological indices. We give some background to these mathe-rnmatical models, and show their applications, which are mainly in chemistryrnand pharmacology. To complete the chapter we present some known resultsrnrnwhich will be relevant to our work.rnChapter 2 focuses on the most common distance-based indices of completernm-ary trees of a given height. We present exact values of all importantrndistance-based indices for complete m-ary trees of a given height. We solvernrecurrence relations to obtain the value of the most well-known index calledrnthe Wiener index. New methods are used to express the other indices (therndegree distance, the eccentric distance sum, the Gutman index, the edge-rnWiener index, the hyper-Wiener index and the edge-hyper-Wiener index) asrnwell. Values of distance-based indices for complete binary trees are corollariesrnof the main results.rnChapter 3 focuses on the general eccentric connectivity index of trees. Wernobtain lower and upper bounds on the general eccentric connectivity indexrnfor trees of given order, trees of given order and diameter, and trees of givenrnorder and number of pendant vertices. The upper bounds for trees of givenrnorder and diameter, and trees of given order and number of pendant verticesrnhold for _ > 1. All the other bounds are valid for 0 < _ _ 1 or 0 < _ < 1.rnWe present all the extremal graphs, which means that our bounds are bestrnpossible.rnChapter 4 focuses on the general eccentric connectivity index of unicyclicrngraphs. We obtain sharp lower and upper bounds on the general eccentricrnconnectivity index for unicyclic graphs of given order, and unicyclic graphsrnof given order and girth. All the lower bounds are valid for 0 < _ < 1 andrnall the upper bounds are valid for 0 < _ _ 1.rnWe use combinatorial methods, algebraic methods, analytic methods andrnvarious graph theoretical techniques, which, in combination with our ideasrnyield new results.