On mixed metric dimension of polycyclic aromatic hydrocarbon networks

Sharma, Sunny Kumar; Bhat, Vijay Kumar; Raza, Hassan; Sharma, Sahil

doi:10.1007/s11696-022-02151-x

Cited by 10 publications

(3 citation statements)

References 42 publications

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“…The computation of metric dimension for molecular graph families is always a challenging task because deciding/selecting landmark vertices (atoms) in minimum numbers is not that easy due to the complexity and scalability of the structures [30]. Although, several authors studied metric dimension and its related variants for several distinct complex molecular graphs as well as for various other graph-theoretic aspects, for example, the metric dimension of H-Napthalenic nanotubes as well as VC 5 C 7 were investigated in [32]; edge metric basis and the dimension of one-pentagonal and one-heptagonal carbon nanocone was investigated [9,20] respectively; for cellulose networks an estimated upper bounds were provided for the cardinality of their resolving sets [33]; the metric dimension of Alpha & 2D lattice boron nanotubes were reported [34]; for polycyclic aromatic hydrocarbons several variants of metric dimension were investigated [35,36]; for chemical graphs of starphene and hollow coronoid several metric dimension variants were reported [37][38][39]. For certain chemical as well as planar graphs, these variants are investigated [20,27,28].…”

Section: Open Accessmentioning

confidence: 99%

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Al-Qudah,

Jaradat,

Sharma

et al. 2024

Phys. Scr.

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For an undirected connected graph $G=G(V,E)$ with vertex set $V(G)$ and edge set $E(G)$, a subset $R$ of $V$ is said to be a resolving in $G$, if each pair of vertices (say $a$ and $b$; $a\neq b$) in $G$ satisfy the relation $d(a, k)\neq d(b, k)$, for at least one member $k$ in $R$. The minimum set $R$ with this resolving property is said to be a metric basis for $G$, and the cardinality of such set $R$, is referred to as the metric dimension of $G$, denoted by $dim_v(G)$. In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by $HCN_{s}$) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of $HCN_{s}$. Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.

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Section: Open Accessmentioning

confidence: 99%

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Al-Qudah,

Jaradat,

Sharma

et al. 2024

Phys. Scr.

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“…Additionally, they established the foundation for these graphs and the presentation of resolving vectors in a comprehensive closed form, specifically in relation to the basis. Sharma et al [25] examined the resolvability parameter, namely the mixed metric dimension, for the intricate molecular structure of the Polycyclic aromatic hydrocarbons network. Additionally, they demonstrated that the mixed metric dimension of the Polycyclic aromatic hydrocarbons network is not limited and does not remain constant.…”

Section: Introductionmentioning

confidence: 99%

Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications

Sardar,

Xu,

Cancan

et al. 2024

J. Comb. Math. Comb. Comput.

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Consider the simple connected graph G with vertex set V(G) and edge set E(G). A graph $G$ can be resolved by $R$ if each vertex’s representation of distances to the other vertices in $R$ uniquely identifies it. The minimum cardinality of the set $R$ is the metric dimension of $G$. The length of the shortest path between any two vertices, x, y in V(G), is signified by the distance symbol d(x, y). An ordered k-tuple $r(x/R)=(d(x,z_1),d(x,\ z_2),…,d(x,z_k))$ represents representation of $x$ with respect to $R$ for an ordered subset $R={\{z}_1,z_2,z_3…,z_k\}$ of vertices and vertex $x$ in a connected graph. Metric dimension is used in a wide range of contexts where connection, distance, and connectedness are essential factors. It facilitates understanding the structure and dynamics of complex networks and problems relating to robotics network design, navigation, optimization, and facility location. Robots can optimize their localization and navigation methods using a small number of reference sites due to the pertinent idea of metric dimension. As a result, many robotic applications, such as collaborative robotics, autonomous navigation, and environment mapping, are more accurate, efficient, and resilient. A claw-free cubic graph (CCG) is one in which no induced subgraph is a claw. CCG proves helpful in various fields, including optimization, network design, and algorithm development. They offer intriguing structural and algorithmic properties. Developing algorithms and results for claw-free graphs frequently has applications in solving of challenging real-world situations. The metric dimension of a couple of claw-free cubic graphs (CCG), a string of diamonds (SOD), and a ring of diamonds (ROD) will be determined in this work.

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“…When analyzing whether a compound's characteristics are responsible for its pharmacological activity, this comparison statement is important [2,3]. Sharma et al [16] studied the mixed metric resolvability of chemical compound polycyclic aromatic hydrocarbon networks. The problem of computing the edge metric dimension of CCS(n) and certain generalisations of these graphs is discussed in this work.…”

Section: Introductionmentioning

confidence: 99%

Edge Resolvability of Crystal Cubic Carbon Structure

Sharma¹,

Bhat²,

Lal³

2022

Preprint

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Chemical graph theory is commonly used to analyse and comprehend chemical structures and networks, as well as their features. The resolvability parameters for graph G= (V, E) are a relatively new advanced field in which the complete structure is built so that each vertex (atom) or edge (bond) represents a distinct position. In this article, we study the resolvability parameters i.e., edge resolvability of chemical graph of crystal structure of cubic carbon CCS(n).

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On mixed metric dimension of polycyclic aromatic hydrocarbon networks

Cited by 10 publications

References 42 publications

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Mathematical analysis of the structure of one-heptagonal carbon nanocone in terms of its basis and dimension

Computing Metric Dimension of Two Types of Claw-free Cubic Graphs with Applications

Edge Resolvability of Crystal Cubic Carbon Structure

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