In this article we introduce the zero-divisor graphs ?P(X) and ?P? (X) of two rings CP(X) CP? (X); here P is an ideal closed sets in X aggregate those functions C(X), whose support lie on P. analogue ring C?(X). We determine when weakly graph W?P(X) coincides with ?P(X). find out conditions topology X, under-which (respectively, (X)) becomes triangulated/ hypertriangulated. realize that a compl...

This work aims to introduce and study a new kind of divisor graph which is called idempotent graph, it denoted by . Two non-zero distinct vertices v1 v2 are adjacent if only , for some non-unit element We establish fundamental properties as well it’s connection with also planarity this graph.

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.

In a manner analogous to a commutative ring, the idealbased zero-divisor graph of a commutative semiring R can be defined as the undirected graph ΓI(R) for some ideal I of R. The properties and possible structures of the graph ΓI (R) are studied.

The algebraic polynomial plays a significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. index is utilized as tool study quantitative structure activity relationship (QSAR) structures property (QSPR) which correlate molecular its different properties activities. Graphs containing finite commutati...

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {x ∈ R \ I| xy ∈ I for some y ∈ R \ I} with distinct vertices x and y adjacent if and only if xy ∈ I. In the case I = 0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literat...