Complement of the generalized total graph

Abstract

Let R be a commutative ring with nonzero identity, H a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (simple) graph GT_{H}(R) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + y is in H. The complement of the generalized total graph $\overline{GT_{H}(R)}$ of $R$ is the (simple) graph with vertex set $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $x + y \notin H.$ In this paper, we investigate certain domination properties of $\overline{GT_{H}(R)}.$ In particular, we obtain the domination number, independence number and a characterization for $\gamma$-sets in $\overline {GT_{P}(\mathbb{Z}_n)}$ where $P$ is a prime ideal of $\mathbb{Z}_n.$ Further, we discuss about properties like Eulerian, Hamiltonian, planarity, chromatic number, clique and girth for $\overline {GT_{P}(\mathbb{Z}_n)}.$