Genus two nilpotent graphs of finite commutative rings

Abstract

Let [Formula: see text] be a finite commutative ring and [Formula: see text] is nilpotent for some [Formula: see text]. The nilpotent graph [Formula: see text] of [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is nilpotent. In this paper, we observe a relationship between zerodivisor graphs, essential graphs and nilpotent graphs of [Formula: see text]. In continuation of genus characterizations in [T. Asir K. Mano and T. Tamizh Chelvam, Correction to: Classification of non-local rings with genus two zerodivisor graphs, Soft Comput. 25 (2021) 3355–3356; K. Selvakumar and M. Subajini, Finite commutative ring with genus two essential 10 graph, J. Algebra Appl. 16(2) (2018) 1850121], we classify all finite commutative rings (up to isomorphism) whose nilpotent graphs are of genus two.