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Graph theory has become a hot topic in Mathematics due to the gradual research done in graph theory. Product of graphs enables the combination or decomposition of its elemental structures. In graph theory there are four standard products, each with its own set of applications and theoretical interpretations. In this article, we study these graph products of zero-divisor graphs of commutative rings and determine their structural properties such as connectivity, diameter and girth. We also determine when the graph product of zero-divisor graphs of ring Zn and Zm are Eulerian. [ABSTRACT FROM AUTHOR]

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) ∈ N(R) if and only if a ∈ N(R) for a ∈ R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.

Chemical graph theory is an extension of mathematical chemistry that explores chemical phenomena and entities using the conceptual frameworks of graph theory. Chemical graph theory, in particular, uses chemical graphs to represent molecular structures. This chemical graph represents bonds and atoms, respectively, as edges and vertices. Cheminformatics predominantly uses chemical graphs as an essential data type for depicting chemical structures. The computable features of graphs lay the groundwork for (quantitative) structure-property and structure-activity predictions, which is an important component of cheminformatics. The physical and chemical characteristics of chemical compounds are thus reflected in these graphs, which can subsequently be reduced to graph-theoretical descriptors or indices. One of the most extensively researched distance-based graph descriptors is the resolving set W (metric dimension), which differentiates each pair of distinct vertices in every connected simple graph. The mixed metric dimension, which is the most significant variant of metric dimension, is determined for the complex molecular graph of a one-pentagonal carbon nanocone (PCN) in this manuscript. We show that only three distinct non-neighboring vertices (least possible requirement) from PCN can be adopted to uniquely identify all of the edges as well as vertices in PCN. [ABSTRACT FROM AUTHOR]

Chemical graph theory is an extension of mathematical chemistry that explores chemical phenomena and entities using the conceptual frameworks of graph theory. In particular, chemical graphs are used to represent molecular structures in chemical graph theory. Edges and vertices in this chemical graph substitute for bonds and atoms, respectively. The primary data types used throughout cheminformatics to depict chemical structures are chemical graphs. The basis for (quantitative) structure-property and structure-activity predictions, a central area of cheminformatics, is laid by the computable properties of graphs. The physical characteristics of molecules are thus reflected in these graphs, which can subsequently be reduced to graph-theoretical indices or descriptors. The resolving set Z , which distinguish every pair of distinct vertices in a connected simple graph, is one of the most well-studied distance-based graph descriptors. In this manuscript, we consider the most significant variation of a metric dimension known as partition dimension and determine it for the molecular complex graph of one-pentagonal carbon nanocone (PCN). We demonstrate that all of the vertices and edges in PCN can be uniquely identified only by considering three distinct non-neighboring partition sets (minimal requirement) from PCN. [ABSTRACT FROM AUTHOR]

Let H = (V , E) be a connected graph of size n. If W is an ordered subset of distinct vertices in H then the subset W is said to be a resolving set for H , if all of the vertices in the graph can be uniquely defined by the vector of distances to the vertices in W. A resolving set W with minimum possible vertices is said to be a metric basis for H. The cardinality of the metric basis is called the metric dimension of the graph H. In this paper, we demonstrate that the metric dimension for some families of related planar graphs is three. [ABSTRACT FROM AUTHOR]

Let ζ = (V;E) be a nth order connected graph. If the distance vectors to the vertices in an ordered subset G of vertices can uniquely identify each vertex of the graph ζ, then the set G is known as resolving set for the graph ζ. The resolving set G with smallest cardinality serves as the metric basis of graph ζ and the cardinality of this smallest resolving set serves as metric dimension for ζ. In this article, two families of convex polytopes that are closely linked are demonstrated and it is found that the metric dimension is three for both the families. [ABSTRACT FROM AUTHOR]

Let H = H(V, E) be a graph. A subset of vertices M in V (H) is said to be a resolving set (or metric generator) for H if every y, z ∈ V (H) with y ≠ z, there exists a vertex a ∈ M such that d(a, y) ≠ d(a, z). A metric generator containing a minimum number of vertices is called a metric basis for H and the cardinality of this metric basis is the metric dimension of H, denoted by dim(H). Let C q n(1, 2) be a graph obtained from the circulant graph Cn(1, 2) by joining n-paths of length q at each vertex of the graph Cn(1, 2). In this work, we show that the metric dimension of the graph C q n(1, 2) is three when n ≡ 0, 2, 3 mod(4) and four when n ≡ 1 mod(4). [ABSTRACT FROM AUTHOR]

Dominant local metric dimension (DLMD) combines two interesting graph theory concepts: domination and local metric dimension (LMD). Graph invariants are a fantastic tool for analyzing graph abstract structures. Metric dimension and DLMD as graph invariants have a wide range of applications, which includes robot navigation, pharmaceutical chemistry, pattern recognition, and image processing. In this paper, we computed the DLMD of the prism graph Y n , antiprism graph A n , and the sunflower graph S F n . [ABSTRACT FROM AUTHOR]

In the past two decades, boron nanotubes have received significant attention from researchers and scientists due to their wide-ranging applications in electronics, nanodevices, optical engineering, nanobiotechnology, and cosmetics. These nanotubular structures composed of boron present exceptional electrical and mechanical properties, making them highly potential nanomaterials. In this article, we study the molecular structure of significant classes of boron nanotubes, namely, trihexagonal boron nanotubes, triangular boron nanotubes, and boron-α nanotubes. Furthermore, we calculate various topological indices for these nanotubes, including the augmented Zagreb index, Sombor index, reduced Sombor index, sum-connectivity index, and arithmetic–geometric index. These indices hold substantial importance in assessing the physical, chemical, and biological characteristics of boron nanotubes. [ABSTRACT FROM AUTHOR]

The topological descriptors are vital devices for investigating and quatifying the molecule's topology. In this paper, we investigate two newly defined ev-degree and ve-degree based topological descriptors for the three significant classes of benzenoid, namely, starphene S T P u , v , w , hollow coronoid H C u , v , w , and hollow coronoid fused with starphene H C S u , v , w . [ABSTRACT FROM AUTHOR]

Discovering a vaccine with reliable and effective treatment for the new corona virus disease 2019 (COVID-19) is indeed a long way off, and there seems to be a critical need to research additional viable medications that could save countless lives in the event of the COVID-19 pandemic. Scientists from all over the world are actively developing medical or anti-virus medications, which are safe and effective for COVID-19. The anti-viral drugs of chloroquine (CQ) and hydroxychloroquine (HCQ) have proved to be an efficient inhibitory effect by preventing the binding of spike covid protein. The determination of a pharmaceutical structure using a topological index allows researchers to have a better understanding of the physiological as well as bio-organic properties of drugs. The goal of this study is to employ molecular graph theory to determine some graph-theoretic parameters related to the molecular graph of CQ and HCQ. In this paper, we present some resolvability parameters such as metric dimension (MD), edge metric dimension (EMD), fault-tolerant metric dimension (FTMD), and fault-tolerant edge metric dimension (FTEMD) of CQ and HCQ. We prove that these resolvability parameters for CQ and HCQ are bounded and constant. [ABSTRACT FROM AUTHOR]