Complement of the generalized total graph of fields

Abstract

Let R be a commutative ring and H be a multiplicative prime subset of R. The generalized total graph is the undirected simple graph with vertex set R and two distinct vertices x and y are adjacent if For a field F, is the only multiplicative prime subset of F and the corresponding generalized total graph is denoted by In this paper, we investigate several graph theoretical properties of where is the complement of the generalized total graph of F. In particular, we characterize all the fields for which is unicyclic, split, chordal, claw-free, perfect and pancyclic.