On some algebraic structures of the graphs

In recent years, graph definitions on algebraic structures, classifications of obtained graph types, and their properties have become interesting research topics concerning the properties of algebraic structures. Consequently, the question of whether graphs can be defined on different elements of frequently used algebraic structures such as groups or rings has emerged. If a graph structure is obtained, to which graph family does it belong? Can the graph parameters, matrices and spectra of the obtained graphs be determined? Can graphs be formed as subgraphs of algebraic structures such as subgroups or subrings? Do similar structures emerge among the graphs of similar algebraic structures? These questions have raised curiosity and led to the formulation of certain graph definitions. Various graph structures defined on rings, such as zero-divisor graphs, nilpotent graphs, nilradical and non-nilradical graphs, total graphs and unit graphs have been introduced in the literature. In this thesis study, new graph structures analogous to nilpotent, total and unit graphs over the ring of integers are defined and their spectral analysis is performed. In the second chapter, fundamental definitions and concepts related to graph parameters and ring parameters are compiled. The third chapter provides information from the literature about zero-divisor, nilpotent, nilradical, non-nilradical, total and unit graph structures and some classifications of these graphs. In the fourth chapter, results regarding the analog graph structures and spectral properties of nilpotent, total and unit graphs constructed over the ring of integers are obtained