Nilpotent graphs with crosscap at most two

Abstract

Let be a commutative ring with identity. The nilpotent graph of , denoted by , is a graph with vertex set , and two vertices and are adjacent if and only if is nilpotent, where . In this paper, we characterize finite rings (up to isomorphism) with identity whose nilpotent graphs can be embedded in the projective plane or Klein bottle. Also, we classify finite rings whose nilpotent graphs are ring graph or outerplanarity index 1,2.

Keywords

• crosscap
• nilpotent
• planar
• outerplanar
• projective