Abstract:

Different spectral characterizations of certain classes of graphs are considered in this dissertation. The large number of papers concerning this topic, indicates that problems of this kind are very interesting in spectral graph theory.
This dissertation, beside Preface and References with 46 items, consists of two chapters: 1. Harmonic graphs, 2. Graphs with maximal index.
Harmonic graphs are introduced and studied in details in Chapter 1. This chapter consists of four sections.
In section 1.1 the definition of harmonic graphs, as well as their basic properties, are given. Harmonic trees are discussed in section 1.2. In section 1.3 we characterize harmonic graphs with small number of cycles; in particular, all unicyclic, bicyclic, tricyclic and tetracyclic graphs are determined. Finally, in section 1.4, we determine all connected 3harmonic graphs with integral spectrum.
The solution of maximal index problem in certain classes of graphs is given in Chapter 2. This chapter consists of four sections.
In sections 2.1 and 2.2 we review some results related to the index of a graph. The emphasis is on graphs with given number of both vertices and edges; in particular we discuss graphs having the fixed number of pendant edges, too. In section 2.3 we give the solution of maximal index problem in the class of connected tricyclic graphs with n vertices and k pendant edges. Finally, in section 2.4, we determine graphs with maximal index among all connected cactuses with n vertices. 