Abstract
Let R be a commutative ring with identity, Z(R) its set of zero-divisors, and H a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the simple undirected graph $$GT_{H}(R)$$ with the vertex set R and two distinct vertices x and y are adjacent if and only if $$x + y \in H.$$ In this paper, we investigate several graph theoretical properties of the complement $$\overline{GT_{H}(R)}.$$ In particular, we obtain a characterization for $$\overline{GT_{P}(R)}$$ to be claw-free or unicyclic or pancyclic. Also, we obtain the clique number and chromatic number of $$\overline{GT_P(R)}$$ and discuss the perfect, planar and outer planarity nature for $$\overline{GT_{P}(R)}.$$ Further, we discuss various domination parameters for $$\overline{GT_{P}(R)}$$ where P is a prime ideal of R.
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