Complement of the generalized total graph of Zn

Abstract

Let R be a commutative ring with identity and H be a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (undirected) simple graph GT H (R) with all elements of R as the vertex set and two distinct vertices x and y are adjacent if and only if x + y ∈ H. The complement of the generalized total graph GT H (R) of R is the (undirected) simple graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y H. In this paper, we investigate certain domination properties of GT H (R). In particular, we obtain the domination number, independence number and a characterization for γ-sets in GT P (Z n) where P is a prime ideal of Z n. Further, we discuss properties like Eulerian, Hamiltonian, planarity, and toroidality of GT P (Z n).