Basic Edge Dominating Functions of Quadratic Residue Cayley Graphs

Abstract

Graph Theory has been realized as one of the most useful branches of Mathematics of recent origin, finding widest applications in all most all branches of sciences, social sciences, and engineering and computer science. Nathanson[8] was the pioneer in introducing the concepts of NumberTheory, particularly, the “Theory of congruences” i n 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. Quadratic residue is an arithmetic function which is defined by: Let p be an odd prime and n, a positive integer such that n 0 (mod p). If the quadratic congruence, ) (mod 2 p n x  has a solution then, n is called a quadratic residue mod p. The Quadratic Residue Cayley graph G(Zp , Q), is the Cayley graph associated with the quadratic residue function. The theory of basic edge dominating functions in Quadratic Residue Cayley Graphs is useful in the selection of modes, those are require to focus on the development of some connected systems like market management, operating system, banking , infrastructure system etc.