Applications on Topological Indices of Zero-Divisor Graph Associated with Commutative Rings

Abstract

A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. Let R be a commutative ring with identity, and Z*(R) is the set of all non-zero zero divisors of R. Then, Γ(R) is said to be a zero-divisor graph if and only if a·b=0, where a,b∈V(Γ(R))=Z*(R) and (a,b)∈E(Γ(R)). We define a∼b if a·b=0 or a=b. Then, ∼ is always reflexive and symmetric, but ∼ is usually not transitive. Then, Γ(R) is a symmetric structure measured by the ∼ in commutative rings. Here, we will draw the zero-divisor graph from commutative rings and discuss topological indices for a zero-divisor graph by vertex eccentricity. In this paper, we will compute the total eccentricity index, eccentric connectivity index, connective eccentric index, eccentricity based on the first and second Zagreb indices, Ediz eccentric connectivity index, and augmented eccentric connectivity index for the zero-divisor graph associated with commutative rings. These will help us understand the characteristics of various symmetric physical structures of finite commutative rings.