The Wiener index and spectrum of the nil clean graph of a finite commutative ring

S. Arumugakaveri; K. Selvakumar; Junaid Nisar

doi:10.46481/jnsps.2026.3679

Authors

S. Arumugakaveri Department of Mathematics, Manonmaniam Sundaranar University College, Affiliated to Manonmaniam Sundaranar University, Govindaperi, Tirunelveli, Tamil Nadu, India
K. Selvakumar selva_158@yahoo.co.in Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu, India
Junaid Nisar Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune 412115, India

Keywords:

Array, Array, Array, Array, Array

Abstract

For a finite commutative ring R, the nil clean graph G_N(R) is the undirected simple graph with the elements of R as its vertex set, where two distinct vertices x and y are joined by an edge exactly when x + y decomposes as a nilpotent element plus an idempotent element -- in other words, when the sum x + y is a nil clean element of R. This article builds upon and extends several results established in Ref. [1]. We introduce the Wiener index of the nil clean graph of a finite commutative ring R, and as an application we determine this index for G_N(mathbb{Z}_n) and for the nil clean graph of finite products of rings of the form mathbb{Z}_n, drawing on the framework of Refs. [1,2]. The paper concludes with a description of the spectrum of the nil clean graph for several classes of rings.

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