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A big amount of vanadium is used in the industry. It is used as a vanadium carbide stabiliser in steel production. Only $ 0.5\% $ 0.5 % vanadium in an iron alloy will double the tensile strength. Vanadium salts are used in leather, ink, dye manufacturing and yellow ceramic glaze. On the other hand, vanadium pentoxide is used as a catalyst, in the chemical industry, for the production of sulphuric acid and in the petroleum refinery industry. Resolvability parameter is the most discussed topic in graph theory. This helps to reorganise a network in a unique way, sometimes in terms of atoms (metric dimension) and sometimes bound (edge metric dimension). We studied resolvability parameters of vanadium carbide network in this article i.e. metric, edge metric and partition dimension. Among these metrics resolvability defined by parameters taking a subset from the set of vertices, if it has a unique representation with all the vertices, then it is called a resolving set. The number of elements in the smallest resolving set is called the metric dimension. While the number of elements of the minimal subset of vertices taking from vertex set is called the edge metric dimension, if the representations are unique of all edges and the set is called the edge resolving set. This study can have an effect in many ways in the research, particularly, with the help of some key points such as network structure analysis, material science via material properties, graph theory applications, predictive modelling, material testing and characterisation, as well as in industrial applications. [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]

Apart from many branches in mathematical sciences, the main share of applications of graph theory nowadays is in chemistry. The graph-theoretical ideas have been used by many chemists for more than two centuries but over the last 30 years, the great significance of graph theory in chemistry has been extensively acknowledged. In a chemical graph, the vertices can symbolize atoms, orbitals, bonds, collection of atoms, molecules, or group of molecules and the edges symbolize the connection between chemical substances and are used to describe reactions, chemical bonds, kinetic models, reaction mechanisms, or other interactions and alterations of the chemical substances. Benzene is the archetypal aromatic molecule, showing exceptional chemical stability as compared with other unsaturated hydrocarbons. Fullerenes are a unique family of carbon-based cage molecules having remarkable properties. Fullerene has potential applications in medicinal chemistry due to its size, hydrophobicity, three-dimensionality, and electronic configurations. In this paper, we will calculate the fault-tolerant partition dimension of classes of fullerene graphs, n-linear benzene graph and partition dimension of polyphenyl chains. [ABSTRACT FROM AUTHOR]

Carbon nanotubes have been actively studied for their applications in filtering of polycyclic aromatic hydrocarbons, sensors, electronics, stretchable devices, and energy storage. The large structures of nanotubes are studied using their induced subgraphs. The partition dimension focusses on the unique distance-based representation of nodes with respect to partition sets such that each node is exclusively identified. The partition and 2 − partition dimension of induced subgraphs of certain nanotubes have been computed in this article. [ABSTRACT FROM AUTHOR]

Let Π = {S1, S2, . . . , Sk} be an ordered partition of the vertex setV(G) of a graphG. The partition representation of a vertexv∈V(G) with respect to Π is thek-tupler(v|Π) = (d(v, S1), d(v, S2), . . . , d(v, Sk)), whered(v, S) is the distance betweenvand a setS. If for every pair of distinct verticesu, v∈V(G), we haver(u|Π) ≠r(v|Π), then Π is a resolving partition and the minimum cardinality of a resolving partition ofV(G) is called the partition dimension ofG. We study the partition dimension of circulant graphs, which are Cayley graphs of cyclic groups. Grigorious et al. [On the partition dimension of circulant graphs] proved thatpd(Cn(1,2, . . . , t)) ≥t+ 1 forn≥ 3. We disprove this statement by showing that ift≥ 4 is even, then there exists an infinite set of values ofn, such that. We also present exact values of the partition dimension of circulant graphs with 3 generators. [ABSTRACT FROM AUTHOR]

Given a connected graph, the distancebetween two verticesis the length of a shortestu−vpath in G. The distancebetween a vertexand a subsetis defined as. An ordered partitionof vertices ofGis a resolving partition ofG, if for any two different verticesu,vofGthere existssuch that. The partition dimension ofGis the minimum number of sets in any resolving partition of G. In this article, we study the partition dimension of the lexicographic product of two graphs. [ABSTRACT FROM AUTHOR]