This article investigates the concept of dominant metric dimensions in zero divisor graphs (ZD-graphs) associated with rings. Consider a finite commutative ring with unity, denoted as , where nonzero elements and are identified as zero divisors if their product results in zero The set of zero divisors in ring is referred to as . To analyze various algebraic properties of , a graph known as the zero-divisor graph is constructed using This manuscript establishes specific general bounds for the dominant metric dimension (Ddim) concerning the ZD-graph of . To achieve this objective, we examine the zero divisor graphs for specific rings, such as the ring of Gaussian integers modulo , denoted as the ring of integers modulo , denoted as , and some quotient polynomial rings. Our research unveils new insights into the structural similarities and differences among commutative rings sharing identical metric dimensions and dominant metric dimensions. Additionally, we present a general result outlining bounds for the dominant metric dimension expressed in terms of the maximum degree, girth, clique number, and diameter of the associated ZD-graphs. Through this exploration, we aim to provide a comprehensive framework for analyzing commutative rings and their associated zero divisor graphs, thereby advancing both theoretical knowledge and practical applications in diverse domains.

Fractional derivatives are notable for various reasons, including the fact that they have properties of non-fractional derivatives, such as making constant derivatives null. Furthermore, nowadays, ...

In the contemporary technological landscape, ensuring confidentiality is a paramount concern addressed through various skillsets. Cryptography stands out as a scientific methodology for safeguarding communication against unauthorized access. Within the realm of cryptography, numerous encryption algorithms have been developed to enhance data security. Recognizing the imperative for nonstandard encryption algorithms to counter traditional attacks, this paper puts forth novel encryption techniques. These methods leverage special corona graphs, star graphs, and complete bipartite graphs, incorporating certain algebraic properties to bolster the secure transmission of messages. The introduction of these proposed encryption schemes aims to elevate the level of security in confidential communication, some of the applications of these schemes are given in the later section.

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs and gives us a new graph obtained by taking one copy of and copies of and joining the th vertex of to every vertex in the th copy of . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products , , , , and and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring $R$ and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set $Z\left(R_E\right)\backslash \left\{\right[0\left]\right\}=R_E\backslash \left\{\right[0],[1\left]\right\}$, where $R_E=\left\{\right[x] :x\in R\}$ and $\left[x\right]=\{y\in R : ann(x)=ann(y\left)\right\}$ is called a compressed zero-divisor graph, denoted by ${\Gamma }_E\left(R\right)$. An edge is formed between two vertices $\left[x\right]$ and $\left[y\right]$ of $Z\left(R_E\right)$ if and only if $\left[x\right]\left[y\right]=\left[xy\right]=\left[0\right],$ that is, iff $xy=0$. For a ring $R$, graph $G$ is said to be realizable as ${\Gamma }_E\left(R\right)$ if $G$ is isomorphic to ${\Gamma }_E\left(R\right)$. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.