On the genus of some total graphs

Abstract

Let be a commutative ring with a proper ideal . A generalization of total graph is introduced and investigated. It is the (undirected) graph with all elements of as vertices, that two distinct vertices are adjacent if and only if where for some and is a multiplicatively closed subset of . This version of total graph is denoted by . We in addition characterize certain lower and upper bounds for the genus of the total graph, and compute genus on finite ring , with respect to some special ideal .

Keywords

• commutative rings
• multiplicatively closed subset
• total graph
• genus