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Minimum resolving sets (edge or vertex) have become integral to computer science, molecular topology, and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for uniquely identifying each item in the network. The metric(respectively edge metric) dimension of a graph is the smallest number of the nodes needed to determine all other nodes (resp. edges) based on shortest path distances uniquely. Metric and edge metric dimensions as graph invariants have numerous applications, including robot navigation, pharmaceutical chemistry, canonically labeling graphs, and embedding symbolic data in low-dimensional Euclidean spaces. A honeycomb torus network can be obtained by joining pairs of nodes of degree two of the honeycomb mesh. Honeycomb torus has recently gained recognition as an attractive alternative to existing torus interconnection networks in parallel and distributed applications. In this article, we will discuss the Honeycomb Rhombic torus graph on the basis of edge metric dimension. [ABSTRACT FROM AUTHOR]

The resolvability parameter is an essential component, especially in the context of network research, due to its theoretical and practical significance. Its importance is evident in several applications and outcomes, including social network analysis, network security, facility location and site selection, and effective routing. We introduce a novel resolvability parameter, Fault-Tolerant Mixed Metric Dimension, in this paper, and this defined as let R m , f be a set that nodes on a graph as both an edge-resolving set and a resolving set. If R m , f can uniquely represent the graph's edges and vertices, then it is referred to as a mixed resolving set, and its all subsets cardinality is called the mixed metric dimension. If all of the graph's vertices and edges are uniquely represented by R m , f ′ , and all subsets of R m , f ′ with of cardinality one less than R m , f likewise have unique representations for all of the graph's vertices and edges, then R m , f is referred to as a Fault-Tolerant Mixed Resolving Set, and If two such sets R m , f 1 and R m , f 2 exist such that R m , f 1 ∩ R m , f 2 ≠ 0 then we say that the graph has exchange property. R m , f 's minimum cardinality is known as its fault-tolerant mixed Metric Dimension. These definitions offer a means of measuring a collection of vertices' capacity to represent graph structures uniquely, taking fault-tolerant and resolution into account. Furthermore, a problem related to the lab's system network is also discussed and linked with this topic in this work. Like a lab engineer is embarking on the creation of a new circular lab, intending to establish where and how many devices within it to supply internet with wire to all systems. A solution to this problem is proving this novel topic authenticity. [ABSTRACT FROM AUTHOR]

Locating the origin of diffusion in complex networks is an interesting but challenging task. It is crucial for anticipating and constraining the epidemic risks. Source localization has been considered under many feasible models. In this paper, we study the localization problem in some path-related graphs and study the edge metric dimension. A subset LE⊆VG is known as an edge metric generator for G if, for any two distinct edges e1,e2∈E, there exists a vertex a⊆LE such that d(e1,a)≠d(e2,a). An edge metric generator that contains the minimum number of vertices is termed an edge metric basis for G, and the number of vertices in such a basis is called the edge metric dimension, denoted by dime(G). An edge metric generator with the fewest vertices is called an edge metric basis for G. The number of vertices in such a basis is the edge metric dimension, represented as dime(G). In this paper, the edge metric dimension of some path-related graphs is computed, namely, the middle graph of path M(Pn) and the splitting graph of path S(Pn).

When there is a difference in the distance between two vertices in a simple linked graph, then a vertex x resolves both u and v. If at least one vertex in S distinguishes each pair of distinct vertices in G, then a set S of vertices in G is referred to as a resolving set. G’s metric dimension is the minimum number of vertices required in a resolving set. A subset S of vertices in a simple connected graph is called an edge metric generator if each vertex can tell any two distinct edges e1 and e2 apart by their respective distances from each other. The edge metric dimension (EMD), denoted as dime(G), is the smallest cardinality of such a subset S that serves as an edge metric generator for G. The primary objective of this study is to investigate the edge metric dimension (EMD) of hexagonal boron nitride and carbon nanotube structures. [ABSTRACT FROM AUTHOR]

The edge metric dimension edim(G) of a connected graph G is the minimum cardinality of a set S of vertices such that each edge is uniquely determined by its distance from the vertices of the set S. In this work, the edge metric dimension of the prism over a graph G (G ... K2), cylinder graphs (Cm ...Pn) and torus graphs (Cm ...Cn) are determined. [ABSTRACT FROM AUTHOR]

The cactus graph has many practical applications, particularly in radio communication systems. Let G = (V, E) be a finite, undirected, and simple connected graph, then the edge metric dimension of G is the minimum cardinality of the edge metric generator for G (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices Ge = fg1, g2, ..., gkg of a connected graph G, for any edge e ∈ E, we referred to the k-vector (ordered k-tuple), r(ejGe) = (d(e, g1), d(e, g2), ..., d(e, gk)) as the edge metric representation of e with respect to Ge. In this regard, Ge is an edge metric generator for G if, and only if, for every pair of distinct edges e1, e2 2 E implies r(e1|Ge), r(e2|Ge). In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: C(n, c, r) and C(n,m, c, r) in terms of the number of cycles (c) and the number of paths (r). [ABSTRACT FROM AUTHOR]

The cactus graph has many practical applications, particularly in radio communication systems. Let $ G = (V, E) $ be a finite, undirected, and simple connected graph, then the edge metric dimension of $ G $ is the minimum cardinality of the edge metric generator for $ G $ (an ordered set of vertices that uniquely determines each pair of distinct edges in terms of distance vectors). Given an ordered set of vertices $ \mathcal{G}_e = \{g_1, g_2, ..., g_k \} $ of a connected graph $ G $, for any edge $ e\in E $, we referred to the $ k $-vector (ordered $ k $-tuple), $ r(e|\mathcal{G}_e) = (d(e, g_1), d(e, g_2), ..., d(e, g_k)) $ as the edge metric representation of $ e $ with respect to $ G_e $. In this regard, $ \mathcal{G}_e $ is an edge metric generator for $ G $ if, and only if, for every pair of distinct edges $ e_1, e_2 \in E $ implies $ r (e_1 |\mathcal{G}_e) \neq r (e_2 |\mathcal{G}_e) $. In this paper, we investigated another class of cacti different from the cacti studied in previous literature. We determined the edge metric dimension of the following cacti: $ \mathfrak{C}(n, c, r) $ and $ \mathfrak{C}(n, m, c, r) $ in terms of the number of cycles $ (c) $ and the number of paths $ (r) $.

Abstract In the modern digital sphere, graph theory is a significant field of research that has a great deal of significance. It finds widespread application in computer science, robotic directions, and chemistry. Additionally, graph theory is used in robot network localization, computer network problems and the formation of various chemical structures for networks. Moreover, it finds uses in exploring diffusion mechanisms and scheduling aircraft as well. The present research project examines and concentrates on the edge version of metric dimension of the Concealed Non-Kekuléan Benzenoid Hydrocarbon, Polythiophene, Backbone DNA network and Bakelite networks. All the mentioned networks have constant edge metric dimension except Bakelite network, as demonstrated by the results. If we talk about the applications of these networks, Polythiophene are used to treat prion disorders. It is also capable of detecting metal ions. The concept of Bakelite, which finds applications in the jewelry, electrical, cookware, sports, and clock industries, had an impact on the invention of modern polymers. The functions of DNA include information encoding, replication, mutation, and recombination gene expression.

If a subset of vertices of a graph, designed in such a way that the remaining vertices have unique identification (usually called representations) with respect to the selected subset, then this subset is named as a metric basis (or resolving set). The minimum count of the elements of this subset is called as metric dimension. This concept opens the gate for different new parameters, like fault-tolerant metric dimension, in which the failure of any member of the designed subset is tolerated and the remaining subset fulfills the requirements of the resolving set. In the pattern of the resolving sets, a concept was introduced where the representations of edges must be unique instead of vertices. This concept was called the edge metric dimension, and this as well as the previously mentioned concepts belong to the idea of resolvability parameters in graph theory. In this paper, we find all the above resolving parametric sets of a convex polytope $ {F}_{♃} $ and compare their cardinalities.

In the modern digital sphere, graph theory is a significant field of research that has a great deal of significance. It finds widespread application in computer science, robotic directions, and chemistry. Additionally, graph theory is used in robot network localization, computer network problems and the formation of various chemical structures for networks. Moreover, it finds uses in exploring diffusion mechanisms and scheduling aircraft as well. The present research project examines and concentrates on the edge version of metric dimension of the Concealed Non-Kekuléan Benzenoid Hydrocarbon, Polythiophene, Backbone DNA network and Bakelite networks. All the mentioned networks have constant edge metric dimension except Bakelite network, as demonstrated by the results. If we talk about the applications of these networks, Polythiophene are used to treat prion disorders. It is also capable of detecting metal ions. The concept of Bakelite, which finds applications in the jewelry, electrical, cookware, sports, and clock industries, had an impact on the invention of modern polymers. The functions of DNA include information encoding, replication, mutation, and recombination gene expression. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph where V is the set of vertices of G and E is the set of edges of G. The distance from the vertex w ∈ V to the edge  e = u v ∈ E is given by d (e , w) = min { d (u , w) , d (v , w) }. A subset S e ⊂ V is called an edge metric generator for G if for every two distinct edges e 1 , e 2 ∈ E , there exists a vertex w ∈ S such that d (e 1 , w) ≠ d (e 2 , w). The edge metric generator with the minimum number of vertices is called an edge metric basis for G and the cardinality of the edge metric basis is called the edge metric dimension denoted by dim e (G). A subset S m ⊂ V is called a mixed metric generator for G if for every two distinct elements x , y ∈ V ∪ E , there exists a vertex w ∈ S such that d (x , w) ≠ d (y , w). A mixed metric generator containing a minimum number of vertices is called a mixed metric basis for G and the cardinality of a mixed metric basis is called the mixed metric dimension denoted by dim m (G). In this paper, we study the edge metric dimension and the mixed metric dimension of a plane graph  T n . [ABSTRACT FROM AUTHOR]

If a subset of vertices of a graph, designed in such a way that the remaining vertices have unique identification (usually called representations) with respect to the selected subset, then this subset is named as a metric basis (or resolving set). The minimum count of the elements of this subset is called as metric dimension. This concept opens the gate for different new parameters, like faulttolerant metric dimension, in which the failure of any member of the designed subset is tolerated and the remaining subset fulfills the requirements of the resolving set. In the pattern of the resolving sets, a concept was introduced where the representations of edges must be unique instead of vertices. This concept was called the edge metric dimension, and this as well as the previously mentioned concepts belong to the idea of resolvability parameters in graph theory. In this paper, we find all the above resolving parametric sets of a convex polytope F♃ and compare their cardinalities. [ABSTRACT FROM AUTHOR]

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]

The study of the characteristics and topologies of networks, including social, computer and biological networks, is known as graph theory. Numerous disciplines, such as epidemiology, which is the study of the transmission and management of infectious diseases like Covid-19, can benefit from the study of graph theory. Like, the first case of COVID-19 (Coronavirus) was reported in November of 2019; by the third week of April, 1,116,643 verified positive cases had been reported, with around 59,158 fatalities. All the meditational structures are symmetric, if we label to change the labels of graphs in either way the graph becomes symmetric and will be no change to it. So, there are many graph theoretical parameters to study the symmetric meditational structures, like metric and edge metric dimension. A notion known as edge metric dimension or edge metric basis states that with a chosen subset from the vertex set, known as the edge resolving set, the whole edge set of a structure uniquely recognised. This idea of vertices resolvability allows for the distinct identification of COVID antiviral medication structures and aids in the research of the structure's structural characteristics. The chosen resolving set and the computed edge metric dimension will be invariant due to the symmetry of the graphs. Arbidol, malaria, carboxylic, thalidomide and theaflavin are the names of these compounds which are discussed in this work. [ABSTRACT FROM AUTHOR]

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.

Chemical graph theory represents an interdisciplinary field at the intersection of chemistry and mathematics, focusing on the analysis of various chemical structures through graph-theoretical frameworks. This area facilitates the examination of complex and extensive chemical configurations by leveraging graph theory principles. In chemical structure graphs, atoms are depicted as vertices, with edges representing the bonds between them. By employing chemical graph theory, the exploration of diverse drug compounds and intricate molecular structures becomes more manageable. A key concept in this field is the resolvability parameter, a metric derived from graph theory. It stipulates that each vertex in a structure must possess a distinct representation via elected vertices, known as a metric basis, resolving set, or locating set in different scientific contexts. The metric dimension, defined as the minimum number of vertices in the resolving set, plays a crucial role in characterizing the structural properties of chemical graphs. In this study, we discussed the edge metric dimension of structures of benzodiazepines, alprazolam, chlordiazepoxide, clobazam, clonazepam, diazepam, nitrazepam, lolendazam, alferez, which are types of drug that used for the treatment of depression. We also put to use the edge metric dimension in anti-depression chemical structures, which are different types of Benzodiazepine. For application of this topic, we select the hospitals and health centers in any emergency situation. In any chronic disease situation, these chosen health centers provide all medical facilities to other hospitals efficiently only because we assigned each hospital and health center a unique code.

Chemical graph theory is a branch of mathematical chemistry which applies classic graph theory to chemical phenomena and entities. Graphs are extensively used in chemistry to recognise the structures of chemical compounds, with the vertices and edges denoting atoms and bonds respectively. Let Γ = (V , E) be a nth order connected graph. If the distance vectors to the vertices in an ordered subset H r of vertices can uniquely identify each vertex of the graph Γ , then the set H r is known as resolving set for the graph Γ . The resolving set H r with smallest cardinality serves as the metric basis for Γ and this cardinality of the metric basis set is termed as the metric dimension of Γ . A unique representation of chemical structures can be obtained using a resolving set. In this manuscript, the most important variant of metric dimension known as mixed metric dimension is taken into consideration and computed it for most important allotrope of carbon commonly known as crystal cubic carbon. Mixed metric dimension of Crystal Cubic Carbon structure has been computed and a comparison among metric dimension, edge metric dimension and mixed metric dimension has been done, from which we find that these three dimensions are equal. [ABSTRACT FROM AUTHOR]

The generalized Petersen graphs are a type of cubic graph formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. This graph has many interesting graph properties. As a result, it has been widely researched. In this work, the edge metric dimensions of the generalized Petersen graphs GP(2l + 1, l) and GP(2l, l) are explored, and it is shown that the edge metric dimension of GP(2l + 1, l) is equal to its metric dimension. Furthermore, it is proved that the upper bound of the edge metric dimension is the same as the value of the metric dimension for the graph GP(2l, l). [ABSTRACT FROM AUTHOR]

The p-type networks are designed with the help of CVNET at topo group Cluj and also given support by nano studio. Such networks develop new p-type surfaces and also represent the decorations of the surfaces. This patched network is designed by two repeated units. The first one is triphenylene having a Z-pen formula and the second one is triphenylene with A-phe. Furthermore, these decorations are acquired as the result of map operations represented in the CVNET software, while its assembling is conducted with the help of the nano studio program. In the literature, its topology is discussed by Omega polynomials which is an applied graph theory topic. Another most applied topic of graph theory is known as the resolvability parameter. So this article studied the resolvability parameters of patched networks, such as metric dimension, and edge metric dimension. These parameters are defined as a resolving set is a subset of vertices of a graph with a condition that each vertex of that graph has a unique code or representation with respect to the chosen subset. Its minimum cardinality is known as metric dimension, while the edge metric dimension is defined by the minimum count of members in the edge resolving set and this set is defined as according to a chosen subset each edge of a graph has unique representations, then this set is known as edge resolving set. A resolving set is a subset of vertices of a graph with a condition that each edge of that graph has a unique code or representation with respect to the chosen subset. It is minimum cardinality is known as the edge metric dimension.

The chemical graph theory helps in understanding the complex structure of molecules. Researchers can acquire a detailed grasp of the physical science, chemical properties, and bio-organic aspects of pharmaceuticals by computing the resolvability, and topological parameters of a drug design. The resolvability constraints for graph G = (V , E) are a relatively new advanced field in which the entire structure is constructed so that each vertex (atom) or edge (bond) indicates a distinct position. A topological index is a type of molecular descriptor that is effectively used in modelling many physicochemical properties in numerous quantitative structure–property/activity relationship (QSPR/QSAR) studies. In the QSPR investigation, topological indices have been used to forecast the bioactivity of chemical molecules. In the proposed study, the metric, edge metric resolvability, and topological indices of oseltamivir and zanamivir, two extensively used antiviral medicines, are computed. Further, we also discuss linear regression analysis between indices and physiochemical properties of drugs. This theoretical analysis may help the chemist and people working in the pharmaceutical industry to predict the properties of zanamivir and oseltamivir drugs without experimenting. [ABSTRACT FROM AUTHOR]

In this paper extremal values of the difference between several graph invariants related to the metric dimension are studied: Mixed metric dimension, edge metric dimension and strong metric dimension. These non-trivial extremal values are computed over all connected graphs of given order. To obtain such extremal values several techniques are developed. They use functions related to metric dimension graph invariants to obtain lower and/or upper bounds on these extremal values and exact computations when restricting to some specific families of graphs. [ABSTRACT FROM AUTHOR]

Let G = G(V,E) be a non-trivial simple connected graph. The length of the shortest path between two vertices p and q, represented by d(p; q), is called the distance between the vertices p and q. The distance between an edge ε = pq and a vertex r in G is defined as d(ε, r) = min{d(p, r), d(q, r)}. If d(r, p) ≠ d(r, q), then the vertex r is said to distinguish (resolve or recognize) two elements (edges or vertices) p, q ∈ V ∪ E. The minimum cardinality of a subset R (Re) of vertices such that all other vertices (edges) of the graph G are uniquely determined by their distances to the vertices in R (Re) is the metric dimension (edge metric dimension) of a graph G. In this article, we consider a family of pentagonal circular ladder (Pm) and investigate its edge metric dimension. We show that, for Pm the edge metric dimension is strictly greater than its metric dimension. Additionally, we answer a problem raised in the recent past, regarding the edge metric dimension of a family of a planar graph Rm (exists in the literature). [ABSTRACT FROM AUTHOR]

Graphs are useful for analysing the structure models in computer science, operations research, and sociology. The word metric dimension is the basis of the distance function, which has a symmetric property. Moreover, finding the resolving set of a graph is NP-complete, and the possibilities of finding the resolving set are reduced due to the symmetric behaviour of the graph. In this paper, we introduce the idea of the edge-multiset dimension of graphs. A representation of an edge is defined as the multiset of distances between it and the vertices of a set, B ⊆ V (Γ) . If the representation of two different edges is unequal, then B is an edge-multiset resolving a set of Γ. The least possible cardinality of the edge-multiset resolving a set is referred to as the edge-multiset dimension of Γ. This article presents preliminary results, special conditions, and bounds on the edge-multiset dimension of certain graphs. This research provides new insights into structure models in computer science, operations research, and sociology. They could have implications for developing computer algorithms, aircraft scheduling, and species movement between regions. [ABSTRACT FROM AUTHOR]

Due to its superlative physical qualities and its beauty, the diamond is a renowned structure. While the green-colored perimantanes diamondoid is one of a higher diamond structure. Motivated by the structure's applications and usage, we look into the edge-based metric parameters of this structure. In this draft, we have discussed edge metric dimension and their generalizations for the generalized perimantanes diamondoid structure and proved that each parameter depends on the copies of original or base perimantanes diamondoid structure and changes with the parameter λ or its number of copies.

Graph theory plays an important role for modelling and designing chemical structures and complex networks. Chemical graph theory is commonly used to analyse and comprehend chemical structures and networks, as well as their features. In graph theory, a chemical structure can be represented by vertices and edges where vertices denote atoms and edges denote molecular bonds. The concept of resolvability parameters for a graph G= (V, E) is a relatively new advanced field in which the complete structure is built so that each vertex (atom) or edge (bond) represents a distinct position. It plays a very prominent role in analysing the overall symmetry and structural properties of new chemical compounds. The goal of this study is to employ chemical graph theory to determine some graph-related parameters related to chemical graphs of hypothesised carbon allotrope. In this article, we study the resolvability parameters, i.e. edge resolvability of the chemical graph of the crystal structure of cubic carbon (CCS(n)). [ABSTRACT FROM AUTHOR]

For an ordered subset S = {v1,..., vk} of vertices in a connected graph G and an edge e′ of G, the edge metric S-representation of e′ = ab is the vector reG(e′|S) = (dG(e′, v1),..., dG(e′, vk)), where dG(e′, vi) = min{dG(a, vi), dG(b, vi)}. A dominant edge metric generator for G is a vertex cover S of G such that the edges of G have pairwise different edge metric S-representations. A dominant edge metric generator of smallest size of G is called a dominant edge metric basis for G. The size of a dominant edge metric basis of G is denoted by Ddime(G) and is called the dominant edge metric dimension. In this paper, the concept of dominant edge metric dimension (DEMD for short) is introduced and its basic properties are studied. Moreover, NP-hardness of computing DEMD of connected graphs is proved. Furthermore, this invariant is investigated under some graph operations at the end of the paper. [ABSTRACT FROM AUTHOR]

A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound’s atoms and the edges represent bonds between the atoms, and the degree (valence) of every vertex (atom) is not more than four. In this paper, we determine the edge metric basis and edge metric dimension (EMD) of the complex molecular graph of a one-heptagonal carbon nanocone ( $HCN_{7}(q)$ ). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in $HCN_{7}(q)$ , uniquely.

Let G be a finite, simple, and connected graph with the vertex set V and the edge set E. The distance between the vertex u and the edge e = v w is defined as d (u , e) = min { d (u , v) , d (u , w) }. A vertex x distinguishes two edges e 1 , e 2 if d (x , e 1) ≠ d (x , e 2). A subset L e of V is called an edge metric generator for G if every two distinct edges of G are distinguished by some vertex of L e. The minimum cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by d i m e (G). Similarly, a vertex x distinguishes two elements (vertices or edges) u , v ∈ V ∪ E if d (x , u) ≠ d (x , v). A subset L m of V is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are distinguished by some vertex of L m. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by d i m m (G). In this paper, we study the edge metric dimension and mixed metric dimension of planar graph Q n. We prove that the edge metric dimension and the mixed metric dimension of Q n are both finite and do not depend upon the number of vertices. [ABSTRACT FROM AUTHOR]

In a graph G , a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S. The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G. In Sedlar and Škrekovski (0000) we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general. [ABSTRACT FROM AUTHOR]

For a given graph G , the metric and edge metric dimensions of G , dim (G) and edim (G) , are the cardinalities of the smallest possible subsets of vertices in V (G) such that they uniquely identify the vertices and the edges of G , respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph G such that dim (G) = a and edim (G) = b for every pair of integers a , b , where 4 ≤ b < a. For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions. [ABSTRACT FROM AUTHOR]

Graph invariants provide an amazing tool to analyze the abstract structures of graphs. Metric dimension and edge metric dimension as graph invariants have numerous applications, among them are robot navigation, pharmaceutical chemistry, etc. In this article, we compute the metric and edge metric dimension of two classes of windmill graphs such as French windmill graph and Dutch windmill graph, and also certain generalizations of these graphs.

Let G = (V (G) , E (G)) be a connected graph. A set of vertices S ⊆ V (G) is an edge metric generator of G if any pair of edges in G can be distinguished by their distance to a vertex in S. The edge metric dimension edim(G) is the minimum cardinality of an edge metric generator of G. In this paper, we first give a sharp bound on e d i m (G - e) - e d i m (G) for a connected graph G and any edge e ∈ E (G) . On the other hand, we show that the value of e d i m (G) - e d i m (G - e) is unbounded for some graph G and some edge e ∈ E (G) . However, for a unicyclic graph H, we obtain that e d i m (H) - e d i m (H - e) ≤ 1 , where e is an edge of the unique cycle in H. And this conclusion generalizes the result on the edge metric dimension of unicyclic graphs given by Knor et al. Finally, we construct graphs G and H such that both e d i m (G) - e d i m (G - u) and e d i m (H - v) - e d i m (H) can be arbitrarily large, where u ∈ V (G) and v ∈ V (H) . [ABSTRACT FROM AUTHOR]

The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G, which distinguishes all pairs of vertices (respectively edges) in G, and it is denoted by dim (G) (respectively edim (G) ). The upper bounds dim (G) ≤ 2 c (G) − 1 and edim (G) ≤ 2 c (G) − 1 , where c (G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover, all leafless cacti that attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves, and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper, we focus on Θ -graphs, as the most simple 2-connected graphs distinct from the cycle, and show that the the upper bound 2 c (G) − 1 holds for both metric dimensions of Θ -graphs; we characterize all Θ -graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2 c (G) − 1 in the class of leafless graphs besides already known extremal cacti and extremal Θ -graphs mentioned here. [ABSTRACT FROM AUTHOR]

Due to its superlative physical qualities and its beauty, the diamond is a renowned structure. While the green-colored perimantanes diamondoid is one of a higher diamond structure. Motivated by the structure's applications and usage, we look into the edge-based metric parameters of this structure. In this draft, we have discussed edge metric dimension and their generalizations for the generalized perimantanes diamondoid structure and proved that each parameter depends on the copies of original or base perimantanes diamondoid structure and changes with the parameter λ or its number of copies. [ABSTRACT FROM AUTHOR]

The vertex (resp. edge) metric dimension of a connected graph G is the size of a smallest set S ⊆ V (G) which distinguishes all pairs of vertices (resp. edges) in G. In Sedlar and Škrekovski (2021) it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph configurations and prove that it takes the greater of the two possible values if and only if the graph contains at least one of these configurations. One of these configurations is the same for both dimensions, while the other two are specific for each of them. This enables us to establish the exact value of the metric dimensions for a unicyclic graph and also to characterize when each of these two dimensions is greater than the other one. [ABSTRACT FROM AUTHOR]

The problem of characterizing the classes of plane graphs with the bounded metric dimension, edge metric dimension, and fault-tolerant metric dimension is of great interest nowadays. In this paper, we study the metric dimension, the fault-tolerant metric dimension, and the edge metric dimension of a two-fold heptagonal-nonagonal circular ladder (denoted by ℍ n 7 , 9 ). We show that the metric dimension and the edge metric dimension of ℍ n 7 , 9 are the same. We also study its fault-tolerant metric dimension and prove that the metric basis and the edge metric basis sets are independent. [ABSTRACT FROM AUTHOR]

Wireless Sensor Network (WSN) is considered to be one of the fundamental technologies employed in the Internet of things (IoT); hence, enabling diverse applications for carrying out real-time observations. Robot navigation in such networks was the main motivation for the introduction of the concept of landmarks. A robot can identify its own location by sending signals to obtain the distances between itself and the landmarks. Considering networks to be a type of graph, this concept was redefined as metric dimension of a graph which is the minimum number of nodes needed to identify all the nodes of the graph. This idea was extended to the concept of edge metric dimension of a graph G, which is the minimum number of nodes needed in a graph to uniquely identify each edge of the network. Regular plane networks can be easily constructed by repeating regular polygons. This design is of extreme importance as it yields high overall performance; hence, it can be used in various networking and IoT domains. The honeycomb and the hexagonal networks are two such popular mesh-derived parallel networks. In this paper, it is proved that the minimum landmarks required for the honeycomb network HC(n), and the hexagonal network HX(n) are 3 and 6 respectively. The bounds for the landmarks required for the hex-derived network HDN1(n) are also proposed. [ABSTRACT FROM AUTHOR]

Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from Geneson (2020). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most k , the maximum possible degeneracy of a graph of metric dimension at most k , the maximum possible chromatic number of a graph of metric dimension at most k , and the maximum n for which there exists a graph of metric dimension at most k that contains K n , n. We also investigate a variant of metric dimension called edge metric dimension and solve another problem from the same paper for n sufficiently large by showing that the edge metric dimension of P n d is d for n ≥ d d − 1 . In addition, we use a probabilistic argument to make progress on another open problem from the same paper by showing that the maximum possible clique number of a graph of edge metric dimension at most k is 2 Θ (k) . We also make progress on a problem from Zubrilina (2018) by finding a family of new triples (x , y , n) for which there exists a graph of metric dimension x , edge metric dimension y , and order n. In particular, we show that for each integer k > 0 , there exist graphs G with metric dimension k , edge metric dimension 3 k (1 − o (1)) , and order 3 k (1 + o (1)). [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., wk} be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE) of e with respect to WE is the k-tuple (d(e, w1), d(e, w2), . . ., d(e, wk)). If distinct edges of G have distinct representation with respect to WE, then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn(1, m) has vertex set {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn(1, 2) and Cn(1, 3) is constant.

Let $G=(V,E)$ be a connected graph of order $n$. $S \subseteq V$ is an edge metric generator of $G$ if any pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of a graph $G$ is the least size of an edge metric generator of $G$. In this paper, we give the characterization of all connected bipartite graphs with $edim=n-2$, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for $edim(G)=n-2$, where $G$ is a graph with maximum degree $n-1$. In addition, the relationship between the edge metric dimension and the clique number of a graph $G$ is investigated by construction.

Minimum resolving sets (edge or vertex) have become an integral part of molecular topology and combinatorial chemistry. Resolving sets for a specific network provide crucial information required for the identification of each item contained in the network, uniquely. The distance between an edge e = cz and a vertex u is defined by d(e, u) = min{d(c, u), d(z, u)}. If d(e1, u) ≠ d(e2, u), then we say that the vertex u resolves (distinguishes) two edges e1 and e2 in a connected graph G. A subset of vertices RE in G is said to be an edge resolving set for G, if for every two distinct edges e1 and e2 in G we have d(e1, u) ≠ d(e2, u) for at least one vertex u ∈ RE. An edge metric basis for G is an edge resolving set with minimum cardinality and this cardinality is called the edge metric dimension edim(G) of G. In this article, we determine the edge metric dimension of one-pentagonal carbon nanocone (1-PCNC). We also show that the edge resolving set for 1-PCNC is independent.

If S = { v 1 , ... , v k } is an ordered subset of vertices of a connected graph G and e is an edge of G, then the vector r G (e | S) = (d G (v 1 , e) , ... , d G (v k , e)) is the edge metric S-representation of e. If the vertices of G have pairwise different edge metric S-representations, then S is an edge metric generator for G. The cardinality of a smallest edge metric generator is the edge metric dimension edim (G) of G. A general sharp upper bound on the edge metric dimension of hierarchical products G (U) ⊓ H is proved. Exact formula is derived for the case when | U | = 1 . An integer linear programming model for computing the edge metric dimension is proposed. Several examples are provided which demonstrate how these two methods can be applied to obtain the edge metric dimensions of some applicable graphs. [ABSTRACT FROM AUTHOR]

A motion of a robot in space is represented by a graph. A robot change its position from point to point and its position can be determined itself by distinct labelled landmarks points. The problem is to determine the minimum number of landmarks to find the unique position of the robot, this phenomena is known as metric dimension. Motivated by this a new modification was introduced by Kelenc. In this paper, we computed the edge metric dimension of barycentric subdivision of Cayley graphs Cay(Zα⊕Zβ), for every α ≥ 6, β ≥ 2 and an observation is made that it has constant edge metric dimension and only three carefully chosen vertices can appropriately suffice to resolve all the edges of barycentric subdivision of Cayley graphs Cay(Zα ⊕ Zβ).

A vertex u ∈ V(G) resolves (distinguish or recognize) two elements (vertices or edges) v, w ∈ E(G)UV(G) if dG(u, v) ≠ dG(u, w) . A subset Lm of vertices in a connected graph G is called a mixed metric generator for G if every two distinct elements (vertices and edges) of G are resolved by some vertex set of Lm. The minimum cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper, we studied the mixed metric dimension for three families of graphs Dn, An, and Rn, known from the literature. We proved that, for Dn the dimm(Dn) = dime(Dn) = dim(Dn), when n is even, and for An the dimm(An) = dime(An), when n is even and odd. The graph Rn has mixed metric dimension 5.

A vertex $k\in V_{G}$ determined two elements (vertices or edges) $\ell,m \in V_{G}\cup E_{G}$ , if $d_{G}(k,\ell)\neq d_{G}(k,m)$ . A set $R_ {\text {m}}$ of vertices in a graph $G$ is a mixed metric generator for $G$ , if two distinct elements (vertices or edges) are determined by some vertex set of $R_ {\text {m}}$ . The least number of elements in the vertex set of $R_ {\text {m}}$ is known as mixed metric dimension, and denoted as $dim_{m}(G)$ . In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are $P^{2}_{n}$ the square of a path, $T(P_{n})$ total graph of a path, the middle graph of a path $M(P_{n})$ , and splitting graph of a path $S(P_{n})$ . We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path $S(P_{n})$ .