Genus of total graphs from rings: A survey

Abstract

Let R be a commutative ring. The total graph T Γ ( R ) of R is the undirected graph with vertex set R and two distinct vertices x and y are adjacent if x + y is a zero divisor in R . In this paper, we present a survey of results on the genus of T Γ ( R ) and three of its generalizations. Keywords: Commutative ring, Total graph, Cayley graph, Genus, Planar