Total graph associated to Γ−semigroup of Zn

Abstract

In this paper we have considered two sets Zn (set of integers modulo n) and Γ = Un (set of unit elements of Zn) so that Zn forms a Γ−semigroup with respect to Un. Now a new graph structure associated to this Γ−semigroup can be introduced by taking all the elements of Zn as the vertices of the graph and any two distinct vertices x and y of Zn are adjacent if there exist an α in Un such that x + α + y ≡ 0 (mod n). This definition is a slight modification of the concept of total graph defined by [2]. We call this graph structure the total graph associated to Γ−semigroup (Zn(Γ)) and denote it by G(Zn(Γ)). This paper focuses on determination of some graphical parameters like the degree of the vertices, number of edges, Girth, Diameter, Planarity and Traversibility of G(Zn(Γ)).