The act of accessing the exact location, or position, of a node in a network is known as the localization of a network. In this methodology, the precise location of each node within a network can be made in the terms of certain chosen nodes in a subset. This subset is known as the locating set and its minimum cardinality is called the locating number of a network. The generalized hexagonal cellular network is a novel structure for the planning and analysis of a network. In this work, we considered conducting the localization of a generalized hexagonal cellular network. Moreover, we determined and proved the exact locating number for this network. Furthermore, in this technique, each node of a generalized hexagonal cellular network can be accessed uniquely. Lastly, we also discussed the generalized version of the locating set and locating number.

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Notes on the Localization of Generalized Hexagonal Cellular Networks

Polyphenyl is used in a variety of applications including high-vacuum devices, optics, and electronics, and in high-temperature and radiation-resistant fluids and greases, it has low volatility, ionizing radiation stability, and high thermal-oxidative properties. The structure of polyphenyls can be represented using a molecular graph, where atoms represent vertices and bonds between atom edges. In a chemical structure, an item/vertex resolves two items if ; similarly, the ordered subset of vertices resolves each pair of distinct vertices named as the resolving set, and its minimum cardinality is described as metric dimension. In the pharmaceutical industry, the competition to find new chemical entities for treating a disease dictates larger project teams that encompass more extensive and diverse synthetic efforts directed at increasingly complicated activity spectra. In this paper, we determine the metric dimension of para-, meta-, and ortho-polyphenyl structures, which are used for structure-activity analysis of these polyphenyl structures.

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Application of Resolvability Technique to Investigate the Different Polyphenyl Structures for Polymer Industry

Let be a simple connected graph. Suppose an - partition of . A partition representation of a vertex is the vector , denoted by . Any partition is referred as resolving partition if such that . The smallest integer is referred as the partition dimension of if the - partition is a resolving partition. In this article, we discuss the partition dimension of kayak paddle graph, cycle graph with chord, and a graph generated by chain of cycles. It has been shown that the partition dimension of the said families of graphs is constant.

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On Partition Dimension of Some Cycle-Related Graphs

Bounds on the Partition Dimension of One Pentagonal Carbon Nanocone Structure

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 metric-based parameters of this structure. In this draft, we have discussed 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> or its number of copies. 

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Vertex Metric-Based Dimension of Generalized Perimantanes Diamondoid Structure

A set of vertices W is a resolving set of a graph G if every two vertices of G have distinct representations of distances with respect to the set W. The number of vertices in a smallest resolving set is called the metric dimension. This invariant has extensive applications in robotics, since the metric dimension can represent the minimum number of landmarks, which uniquely determine the position of a robot moving in a graph space. Finding the metric dimension of a graph is an NP-hard problem. We present exact values of the metric dimension of Kayak Paddles graph and Cycles with chord.

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Computing the metric dimension of Kayak Paddles graph and Cycles with chord

Consider a simple connected undirected graph G = (VG ,EG), where VG represents the vertex set and EG represents the edge set respectively. A subset B of VG is called a resolving set if for every two distinct vertices x, y of G there is a vertex v in set B such that d(x,v) ≠ d(y,v). The resolving set of minimum cardinality is called metric basis of graph G . This minimal cardinality of metric basis is denoted by β(G), and is called metric dimension of G. A subset D of V is called doubly resolving set if for every two vertices x, y of G there are two vertices u, v ∈ D such that d(u,x) -d(u,y) ≠ d(v,x) -d(v,y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by ψ(G).Some partial cases for metric dimension of circulant graph Cn(1,2,3) for n ≥ 12 has been discussed in [21]. Afterwards, problem of finding metric dimension for circulant graph Cn(1,2,3), n ≥ 12 has been completely solved by Borchert et al., in [7].In this paper, we prove that ψ (Cn (1, 2, 3)) = β (Cn (1, 2, 3)) = {4 if n ≠ 1(mod 6), 5 otherwise.

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On Minimal Doubly Resolving Sets of Circulant Graphs

Let G = (V,E) be a simple connected and undirected graph, where V and E represent the vertex and edge set, respectively. The vertices x and y doubly resolve the vertices u and v if the following condition is satisfied d(u, x)− d(u, y) 6= d(v, x)− d(v, y). A subset D of vertex set V of G is said to be doubly resolving set of G if for every pair x′, y′ of distinct vertices of G, there exist two vertices x, y in D which doubly resolve the vertices x′, y′. A minimal doubly resolving set is a doubly resolving set which has minimum cardinality. The cardinality of minimal doubly resolving set is denoted by ψ(G). Let β(G) denotes the metric dimension of graph G which is the cardinality of minimal resolving set, then we have β(G) ≤ ψ(G) since every doubly resolving set is a resolving set, too. Borchert and Gosselin et al. solved the problem of finding metric dimension for Harary graph H4,n, n ≥ 8. In this paper, we find the minimal doubly resolving set, and hence the cardinality ψ(H4,n) for Harary graph H4,n, n ≥ 8.

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Saba Sultan

Papers

Computing the metric dimension of Kayak Paddles graph and Cycles with chord

On Minimal Doubly Resolving Sets of Circulant Graphs

On the minimal doubly resolving sets of Harary graph

Minimal doubly resolving sets of antiprism graphs and Möbius ladders