Search or add a thesis

Advanced Search (Beta)
Home > On Metric Dimension of Some Families of Graph

On Metric Dimension of Some Families of Graph

Thesis Info

Access Option

External Link

Author

Murtaza Ali

Program

PhD

Institute

National University of Computer and Emerging Sciences

City

Islamabad

Province

Islamabad

Country

Pakistan

Thesis Completing Year

2015

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/9993/1/Murtaza%20Ali_Maths_2015_NUCES%2c%20Isb_main%20part.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726799504

Asian Research Index Whatsapp Chanel
Asian Research Index Whatsapp Chanel

Join our Whatsapp Channel to get regular updates.

Similar


For a connected graph G the distance d(u;v) between two vertices u;v 2V(G) is the length of the shortest path between them. A vertex w of a graph G is said to resolve two vertices u and v of G if d(w;u) 6= d(w;v). Let W = fw1;w2; ::::;wkg be an ordered set of vertices of G and let v be a vertex of G. The representation of a vertex v with respect to W denoted by r(vjW) is the k-tuple (d(v;w1);d(v;w2); :::;d(v;wk)). If distinct vertices of G have distinct representations with respect toW, thenW is called a resolving set for G. The metric dimension of G denoted by dim(G), is the minimum cardinality of a resolving set of G. Graph structure can be used to study the various concepts of Navigation in space. A work place can be denoted as vertex in the graph, and edges denote the connections between the places. The problem of minimum machines (or Robots) to be placed at certain vertices to trace each and every vertex exactly once is a classical one. This problem can be solved by using networks where places are interconnected in which, the navigating agent moves from one vertex to another in the network. The places or vertices of a network where we place the machines (robots) are called landmarks. The minimum number of machines required to locate each and every vertex of the network is termed the metric dimension and the set of all minimum possible number of landmarks constitute metric basis. In this thesis, the metric dimension of some well known families of graphs has been investigated. It is shown that the families of graphs obtained from the path graph by the power, middle and total graph operation have a constant metric dimension. We compute the metric dimension of some rotationally symmetric families of graphs and show that only 2 or 3 vertices appropriately chosen suffice to resolve all the vertices of these graphs. In this thesis, we also compute the metric dimension of some families of convex polytopes with pendant edges. It is shown that the metric dimension of these families of graphs is constant and is independent of the order of these graphs. The metric dimension of the splitting graphs of two families of graph has been computed. We prove that the metric dimension of these graphs is unbounded and depends on the order of the corresponding graph.
Loading...
Loading...

Similar Books

Loading...

Similar Chapters

Loading...

Similar News

Loading...

Similar Articles

Loading...

Similar Article Headings

Loading...

یار پرانے چھڈ کے ٹریوں۔۔۔

یار پرانے چھڈ کے ٹریوں نویاں من پسنداں نال
ہتھیں دتیاں ہوئیاں گنڈھاں، بہہ کے کھولیں دنداں نال
ہک پل کول کھلوتیاں میرے، جے کر جگ نے ویکھ لیا
پھیر یقین کسے نہیں کرنا، قسماں تے سوگنداں نال
قدم قدم تے ہے پئی چمدی، منزل پیر مسافر دے
چار دیہاڑے بہہ کے جس نے کڈھے دانش منداں نال
تن من دھن قربان کرایا، دین بچایا نانے دا
شاہ حسینؑ، شہید ہوئے نے، خویشاں تے فرزنداں نال

Scope of State in Legislation from Islamic Perspective

Islamic law is basically a part of a holistic system based primarily on the divine message enclosed in the Holy Qur'an and traditions of the Prophet (SAW), which are the main fundamental sources of Islamic law. After the demise of the Prophet (SAW), field of Ijtehād started development, which was already approved by the Prophet (SAW) in his life. The companions of the Prophet (SAW) developed the notion of Ijmā while early Muslim jurists discovered the Qiyās, Maslaha, Istislāh, Istehsān etc. Determining the location of authority and its scope in law-making has remained a complex situation for the western philosophers since long. As far as the case of Muslims is concerned, they are in the position to find solution of this situation as to where the authority dwells; enabling them to resolve many queries which seemed to be unanswered for the long time. It is important for the Muslims to comprehend the concept of Islamic institutions from the perspective of Islamic frame work and legal as well as constitutional history of Islamic history.

An Adaptive Trust-Aware Brokerage Model for Cross-Cloud Federation

Establishing trust in cloud computing has been a major concern for cloud users since the very beginning of pay-as-you-go service. In the recent years, cross-cloud federation has enabled cloud providers to share or lease resources from each other. Contrary to the hesitation of cloud users for cloud adoption, it is now the cloud providers that are reluctant to take part in federation due to lack of trust on their unknown counterparts. A recent void has been observed to address the challenges of trustworthy resource exchanges within the federation. This research has established that trust awareness among cloud providers requires a comprehensive trust framework that is aligned with the nature of federation. A detailed requirement analysis for trust in cross-cloud federation has been performed in this research. This analysis is based on four founding principles of cloud-to-cloud trust paradigm namely bi-directionality, composite trust, delegation control and resource aware trust evaluation. Afterwards, requirements originated from these principles are aligned with the attributes of trust and cloud federation with the help of a detailed requirement matrix. Keeping in view this requirement matrix, an adaptive trust-aware brokerage model has been developed. This model offers dynamic trust establishment approaches that are a function of relationship among service providers. Three different approaches i.e. Conjunctive Accumulation of Trust (ConAccT), Numerical Accumulation of Trust (NAccT) and Cooperation Threshold Estimation (CTE) are developed as part of the proposed model. ConAccT is based on belief calculus and may be useful in case of highly competitive collaborating scenarios where detailed analysis of trust is required to decide cooperation among Cloud Service Providers (CSPs). NAccT approach is based on numerical calculus and is useful in less competitive scenarios and can be combined with metrics other than trust i.e. performance, availability, resource specifications etc. CTE approach is an extension to NAccT and presents a use case of utilizing the performance metric of a CSP combined with its trust metric to evaluate risk of failure in a collaborative project. The significance of these approaches has been verified by implementing the adaptive trust-aware model as a trusted broker based Clouds4Coordination (C4C) system developed for Architecture/Engineering/Construction (AEC) industry. This C4C system is currently implemented in United Kingdom in collaboration with Cardiff University, UK and Rutgers Discovery Informatics Institute (RDI2), USA. Experimental evaluation of these approaches suggests their suitability in varying scenarios of collaborative computing in construction industry. It has been verified that trust-aware relationships within the federation stays for a longer duration of time during collaborative projects. Moreover, an in depth analysis of proposed approach has shown that trust awareness is beneficial in terms of successful service delivery, earlier project completion and reduction in uncertainty of collaboration. A comparative analysis with state-of-art approaches have demonstrated the efficiency of proposed approaches to identify participants of federation that can cause potential risks and unnecessary delays in the projects.