# Very cost effective bipartition in Gamma(Z_n)

## TL;DR

This paper explores the conditions under which the zero-divisor graph of Z_n and related graphs have very cost effective bipartitions, focusing on cases where n is a product of distinct primes, and extends the analysis to line and total graphs.

## Contribution

It introduces the concept of very cost effective bipartitions in zero-divisor graphs of Z_n and investigates their existence for various n, including line and total graphs.

## Key findings

- Characterization of very cost effective bipartitions for Gamma(Z_n) when n is a product of distinct primes.
- Conditions under which N(Z_n) and Omega(Z_n) graphs have very cost effective bipartitions.
- Results on very cost effective bipartitions of line and total graphs of Gamma(Z_n).

## Abstract

Let Z_n be the finite commutative ring of residue classes modulo n and Gamma(Z_n) be its zero-divisor graph. The nilradical graph and non-nilradical graph of Z_n are denoted by N(Z_n) and Omega(Z_n) respectively. In 2012, Haynes et al. [5] introduced the concept of very cost effective graph. For a graph G = (V,E) and a set of vertices S subset of V, a vertex v in S is said to be very cost effective if it is adjacent to more vertices in V\S than in S. A bipartition Pi = {S, V\S} is called very cost effective if both S and V\S are very cost effective sets [5,6]. In this paper, we investigate the very cost effective bipartition of Gamma(Z_n), where n = p_1 p_2 ... p_m, here all p_i's are distinct primes. In addition, we discuss the cases in which N(Z_n) and Omega(Z_n) graphs have very cost effective bipartition for different n. Finally, we derive some results for very cost effective bipartition of the Line graph and Total graph of Gamma(Z_n), denoted by L(Gamma(Z_n)) and T(Gamma(Z_n)) respectively.