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An e partition X of the vertex set V (H) of a connected graph H is the collection of e number of ordered disjoint subsets of V (H), denoted as X = {X1,X2>,: ::, Xe}. The representation of a vertex u is a distance vector r(ujX) = (d(u,X1), d(u,X2),: ::, d(u,Xe)), where d(u,Xi) is the distance of u from Xi. Any ordered e partition X is referred as resolving partition if representations of all the vertices are distinct. The smallest integer e is referred as the partition dimension of the graph. The advancement in the concept of partition dimension is fault-tolerant partition dimension where the representations are distinct at two places for each pair of vertices. In this paper, we compute the partition dimension and fault-tolerant partition dimension of some planar graphs. [ABSTRACT FROM AUTHOR] more...

The partition dimension of a graph is the minimum number of vertex partitions such that every vertex has different distances to the ordered partitions. Many resolving partitions for trees have all vertices not in an end-path in the same partition. This reduces the problem of the partition dimension of trees into finding the partition dimension of palms, the end-paths from a branch. In this paper, we construct a resolving partition for trees using resolving partitions of their palms. We also study some bounds for the partition dimension of palms and also find the partition dimension of regular palm and olive trees. [ABSTRACT FROM AUTHOR] more...

Partition dimension problems involve dividing a graph's vertex set into a minimum number of disjoint sets so that each vertex is different with respect to the representation from each disjoint set. As a result of the development of this method, a number of applications have arisen in a number of fields such as drug design, navigation of robots, pattern recognition, and image processing. In this paper, we have calculated the partition dimension of oxide and zigzag benzenoid networks, and the subdivision of benzenoid hydrocarbon and triangular benzene networks. [ABSTRACT FROM AUTHOR] more...

Let G = (V, E) be a simple, connected graph with vertex set V(G) and E(G) edge set of G. For two vertices a and b in a graph G, the distance d(a, b) from a to b is the length of shortest path a-b path in G. A k-ordered partition of vertices of G is represented as Rp = {Rp1, Rp2,. . ., Rpk} and the representation r(a|Rp) of a vertex a with respect to Rp is the vector (d(a|Rp1), d(a|Rp2), . . ., d(a|Rpk)). The partition is called a resolving partition of G if r(a|Rp) ≠ r(b|Rp) for all distinct a, b ∈ V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically Tn, Un, Vn, and An, and proved that these graphs have constant partition dimension. [ABSTRACT FROM AUTHOR] more...

Let $ G = (V, E) $ be a simple, connected graph with vertex set $ V(G) $ and $ E(G) $ edge set of $ G $. For two vertices $ a $ and $ b $ in a graph $ G $, the distance $ d(a, b) $ from $ a $ to $ b $ is the length of shortest path $ a-b $ path in $ G $. A $ k $-ordered partition of vertices of $ G $ is represented as $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ is the vector $ (d(a|{R}{p_1}), d(a|{R}{p_2}), \dots, d(a|{R}{p_k})) $. The partition is called a resolving partition of $ G $ if $ r(a|{R}{p}) \ne r(b|{R}{p}) $ for all distinct $ a, b\in V(G) $. The partition dimension of a graph, denoted by $ pd(G) $, is the cardinality of a minimum resolving partition of $ G $. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically $ \mathbb{T}_n $, $ \mathbb{U}_n $, $ \mathbb{V}_n $, and $ \mathbb{A}_n $, and proved that these graphs have constant partition dimension. more...

The notion of partition dimension was initially introduced in the field of graph theory, primarily to examine distances between vertices. The local partition dimension extends this idea by incorporating specific conditions into how vertices are represented. In graph theory, it is customary to represent the partition dimension of a graph as $pd(G)$ . Network localization, on the other hand, is the process of precisely determining the position of nodes within a network concerning a selected subset of nodes, called the locating set. The smallest size of its locating set is used to represent the locating number of a network. The generalized hexagonal cellular network provides an innovative framework for network planning and analysis. In our study, we investigate the partition dimension of a generalized hexagonal cellular network and provide a rigorous proof of its exact partition dimension. Hence, our approach ensures the distinct recognition of each node within a generalized hexagonal cellular network. Additionally, we explore the utilization of the metric dimension in flood relief camping by the National Disaster Management Authority (NDMA) Pakistan during floods in 2022. NDMA established relief camps and relief centers with unique codes to rescue humans and animals. more...

In the aircraft sector, honeycomb composite materials are frequently employed. Recent research has demonstrated the benefits of honeycomb structures in applications involving nanohole arrays in anodized alumina, micro-porous arrays in polymer thin films, activated carbon honeycombs, and photonic band gap honeycomb structures. The resolvability parameter is the area of graph theory that is most commonly explored. This results in an original network reconfiguration. Occasionally in terms of atoms (metric dimension), and sometimes in terms of bounds (edge metric dimension). In this article, we examined the honeycomb rhombic torus’s metric, edge metric, mixed metric, and partition dimension. The application of edge metric dimension is also discussed in it. more...

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] more...

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] more...

In graph theory, a commonly used concept is the partition dimension or partition metric basis is to uniquely identify the node set of a structure by dividing it into smaller subsets, known as partition resolving sets. These subsets can then be used to define the partition dimension or partition metric of the graph. This concept is useful in the analysis and understanding of the structure and properties of graphs. This article describes a partition dimension of the line graph of the honeycomb network, the Aztec diamond network, and the extended Aztec diamond network. [ABSTRACT FROM AUTHOR] more...

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] more...

For every graph G, the dominating partition dimension of G is either the same as its partition dimension or one higher than its partition dimension. In this paper, we consider some general connections among these three graph parameters: partition dimension, locating-chromatic number, and dominating partition dimension. We will show that βp(G)≤ηp(G)≤χL(G) for any graph G with at least 3 vertices. Therefore, we will derive properties for which graphs G have ηp(G)=βp(G) or ηp(G)=βp(G)+1. [ABSTRACT FROM AUTHOR] more...

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] more...

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3rd of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure. more...

The characterizations of all graphs of order n with partition dimension 2; n - 2; n - 1 or n have been completely studied. Recently, all graphs of order n ≥ 11 and diameter two with partition dimension n-3 have been characterized. In this paper, we continue characterizing all graphs on n vertices with partition dimension n-3 and diameter either 3 or 4. This completes the characterization of all graphs of order n ≥ 11 with partition dimension n - 3. [ABSTRACT FROM AUTHOR] more...

Let Λ = {B1, B2, . . ., Bl} be an ordered l-partition of a connected graph G(V (G), E(G)). The partition representation of vertex x with respect to Λ is the l-vector, r(x|Λ) = (d(x, B1), d(x, B2), . . ., d(x, Bl)), where d(x, B) = min{d(x, y)|y ∈ B} is the distance between x and B. If the l - vectors r(x|Λ), for all x ∈ V (G) are distinct then l - partition is called a resolving partition. The least value of l for which there is a resolving l - partition is known as the partition dimension of G symbolized as pd(G). In this paper, the partition dimension of circulant graphs Cn(1, 2, 3, 4) is computed for n ≥ 8 as, pd(Cn(1, 2, 3, 4)) = { if 8 ≤ n ≤ 9; { 6, if n = 10; {5, if n ≥ 11. [ABSTRACT FROM AUTHOR] more...

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings. [ABSTRACT FROM AUTHOR] more...

In this paper, we compute the partition dimension and local metric dimension of the extended annihilating-ideal graph 피픸픾(R) associated to a commutative ring R which is denoted by dimPubscript>(피픸픾 (R)) and diml(피픸픾 (R)) respectively. In addition, we characterize diml (피픸픾 (R)) for direct product of rings and the ring of integers ℤn. [ABSTRACT FROM AUTHOR] more...

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for the multiprocessor interconnection networks. Oxide interconnection networks are extracted from the silicate networks having applications in multiprocessor systems due to their symmetry, smaller diameter, connectivity and simplicity of structure, and a constant number of links per node with the increasing size of the network can avoid overloading of nodes. The fault tolerant partition basis assigns unique addresses to each processing node in terms of distances (hops) from the other subnets in the network which work in the presence of faults. In this manuscript, the partition and fault tolerant partition resolvability of oxide interconnection networks have been studied which include single oxide chain networks ( S O X C N ), rhombus oxide networks ( R H O X N ) and regular triangulene oxide networks ( R T O X N ). Further, an application of fault tolerant partition basis in case of region-based routing in the networks is included. [ABSTRACT FROM AUTHOR] more...

Little is known about the partition dimension of infinite graphs. Tomescu studied graphs where the set of vertices is the set of points of the integer lattice. We generalize these graphs and present several exact values, lower bounds and upper bounds on the partition dimension of infinite graphs. [ABSTRACT FROM AUTHOR] more...

The partition dimension is the most complicated resolving parameter of all the resolvability parameters. In this parameter, the vertex set of a graph is completely subdivided into subsets so that each vertex of a graph has a unique identification in terms of the partitioning of its selected subsets. In this paper, we consider Carbon nanocone structure (CNC) more specifically one pentagonal carbon nanocone CNCζ,5. It is also regarded as a useful structure in terms of various applications. To delve deeper into this structure, we provide a graphical representation of it and identified sharp bounds for the partition dimension. We conclude that the pentagonal carbon nanocone CNCζ,5 has partition dimension ≤4. more...

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings. more...

The partition dimension is the most complicated resolving parameter of all the resolvability parameters. In this parameter, the vertex set of a graph is completely subdivided into subsets so that each vertex of a graph has a unique identification in terms of the partitioning of its selected subsets. In this paper, we consider Carbon nanocone structure (C N C) more specifically one pentagonal carbon nanocone CN C ζ , 5 . It is also regarded as a useful structure in terms of various applications. To delve deeper into this structure, we provide a graphical representation of it and identified sharp bounds for the partition dimension. We conclude that the pentagonal carbon nanocone CN C ζ , 5 has partition dimension ≤ 4. [ABSTRACT FROM AUTHOR] more...

In November 2019, there was the first case of COVID-19 (Coronavirus) recorded, and up to 3$ ^{rd }$ of April 2020, 1,116,643 confirmed positive cases, and around 59,158 dying were recorded. Novel antiviral structures of the SARS-COV-2 virus is discussed in terms of the metric basis of their molecular graph. These structures are named arbidol, chloroquine, hydroxy-chloroquine, thalidomide, and theaflavin. Partition dimension or partition metric basis is a concept in which the whole vertex set of a structure is uniquely identified by developing proper subsets of the entire vertex set and named as partition resolving set. By this concept of vertex-metric resolvability of COVID-19 antiviral drug structures are uniquely identified and helps to study the structural properties of structure. more...