Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the "Theory of Congruences" in Graph Theory, thus paving way for the emergence of a new class of graphs, namely "Arithmetic Graphs". Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph.In this paper, we present some results related to basic properties of direct product graphs of Euler totient Cayley graphs with Arithmetic graph.

Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Using the number theoretic function Euler totient function an Euler totient Cayley graph is defined and in this paper we study the Strong domination parameters of Euler totient Cayley graphs and Arithmetic V n Graphs.

In their recent study of arithmetic graphs associated with certain arithmetic functions, the authors have introduced a new class of arithmetic graphs associated with Mangoldt function Λ(n), n ≥ 1, an integer and studied their basic properties and vertex cover. In this paper the edge cover, edge domination set, bondage number, non -bondage number and their parameters have been obtained for these graphs.

Number Theory is one of the oldest branches of mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern.

Domination in graphs is the fast growing area of research in Graph theory that has emerged rapidly in the last three decades. An introduction and an extensive overview on domination in graphs and related topics is surveyed and detailed in the two books by Haynes et al. [ 6, 7 ]. They have applications in diverse areas such as logistics and networks design, mobile computing, resource allocation and telecommunication etc. Product of graphs occurs naturally in discrete mathematics as tools in combinatorial constructions. They give rise to important classes of graphs and deep structural problems. In this paper the concept of total dominating functions of corona product graph of a cycle with a complete graph is studied and some results on minimal total dominating functions are obtained.