Some Properties of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Abstract

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.

On Vn-Arithmetic Graph

“…Therefore each vertex of V n is connected to some vertex in V n . Some domination parameters and domination parameters of direct product graphs of Cayley graphs with Arithmetic graphs are presented in [6,7,4,8,3,2]. In this paper, we obtain some properties of V n -Arithmetic graph, maximum degree, minimum degree, number of edges, diameter, radius, Hamiltonian and Eulerian.…”

Some Domination Parameters of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

“…In other cases, by the definition of adjacency in there exist edges between prime number vertices and their prime power vertices and also to their prime product vertices. Therefore each vertex of is connected to some vertex in The domination parameters of these graphs are studied by S.Uma Maheswari [8] and we present some of the results which we need without proofs and can be found in [12]. …”

“…The domination parameters of these graphs are studied by Uma Maheswari [8] and we present some of the results which we need without proofs and can be found in [10]. Then the total domination number of is given by where is the length of the longest stretch of consecutive integers in each of which shares a prime factor with…”

“…Then is a completely disconnected graph on vertices [13]. Since there is no edge between these vertices, all these vertices form a dominating set.…”

“…Now we consider the direct product graph of Euler totient Cayley graphs with Arithmetic graphs. The properties of these graphs are presented in [8]. We briefly present Euler totient Cayley graph and Arithmetic graph.…”