scholarly journals Metric Dimension on Path-Related Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Saqib Nazeer ◽  
Muhammad Hussain ◽  
Fatimah Abdulrahman Alrawajeh ◽  
Sultan Almotairi
Keyword(s):  
Fiber Optics ◽  
Robot Navigation ◽  
Vital Role ◽  
Telecommunication Networks ◽  
Location Problems ◽  
Computer Networking ◽  
Metric Dimension ◽  
Facility Location Problems ◽  
Chemical Structures ◽  
Path Graph

Graph theory has a large number of applications in the fields of computer networking, robotics, Loran or sonar models, medical networks, electrical networking, facility location problems, navigation problems etc. It also plays an important role in studying the properties of chemical structures. In the field of telecommunication networks such as CCTV cameras, fiber optics, and cable networking, the metric dimension has a vital role. Metric dimension can help us in minimizing cost, labour, and time in the above discussed networks and in making them more efficient. Resolvability also has applications in tricky games, processing of maps or images, pattern recognitions, and robot navigation. We defined some new graphs and named them s − middle graphs, s -total graphs, symmetrical planar pyramid graph, reflection symmetrical planar pyramid graph, middle tower path graph, and reflection middle tower path graph. In the recent study, metric dimension of these path-related graphs is computed.

scholarly journals Resolvability in Subdivision of Circulant Networks Cn1,k

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Jianxin Wei ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Ghulam Abbas ◽  
Muhammad Imran
Keyword(s):  
Facility Location ◽  
Connected Graph ◽  
Location Problems ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Facility Location Problems ◽  
Wide Range ◽  
Circulant Networks

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

On 2-partition dimension of the circulant graphs

2021 ◽  
pp. 1-11
Author(s):  
Asim Nadeem ◽  
Agha Kashif ◽  
Sohail Zafar ◽  
Zohaib Zahid
Keyword(s):  
Robot Navigation ◽  
Acta Math ◽  
Telecommunication Networks ◽  
Metric Dimension ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Message Delay ◽  
Network Topologies ◽  
Partition Dimension ◽  
High Connectivity

The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, …, Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , …, d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

Integration of AHP-QFD for selecting facility location

2015 ◽  
Vol 22 (3) ◽  
pp. 411-425 ◽  
Author(s):  
Rajesh Chadawada ◽  
Ahmad Sarfaraz ◽  
Kouroush Jenab ◽  
Hamid Pourmohammadi
Keyword(s):  
Facility Location ◽  
Solution Process ◽  
Pairwise Comparison ◽  
Integrated Approach ◽  
Point Of View ◽  
Location Problems ◽  
Analytic Hierarchy ◽  
Content Type ◽  
Facility Location Problems ◽  
Managerial Experience

Purpose – The purpose of this paper is to describe and implements an analytic hierarchy process (AHP)-QFD model for selecting the best location from an organization point of view which picks the site with the best opportunity requirements. Integration of AHP-QFD process gives us a new approach to assist organizations through observing various factors and selecting the best location among different alternatives. This approach uses AHP method to match the preferences required by decision makers and these preferences are applied to the characteristics of QFD. The model fundamental requirement are perfect potential locales and the areas are contrasted and both quantitative and qualitative elements to permit directors to join managerial experience and judgment in the answer process. The AHP-QFD model is also applied on a case study to illustrate the solution process. Design/methodology/approach – The integration of AHP and QFD is used to analyze available options and select the best alternative. This can be done by ranking each criterion through a pairwise comparison. Given collected data, the QFD approach is used to find the capability of each criterion. Findings – Integration of AHP-QFD is used to select the best alternative in facility location. This integrated approach can be best used in dealing with facility location problems. Originality/value – The developed AHP-QFD model in facility location problems, facilitates the inclusion of market criteria and decision maker opinion into the traditional cost function, which has been mainly distance base in the literature.

On Constrained Facility Location Problems

2008 ◽  
Vol 23 (5) ◽  
pp. 740-748 ◽  
Author(s):  
Wei-Lin Li ◽  
Peng Zhang ◽  
Da-Ming Zhu
Keyword(s):  
Facility Location ◽  
Location Problems ◽  
Facility Location Problems
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