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On Irregular Total Labelings

Thesis Info

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Author

Ali Ahmad

Supervisor

Martin Baca

Program

PhD

Country

Pakistan

Thesis Completing Year

2005

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/handle/123456789/614

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726797865

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A labeling of a graph is a map that carries graph elements to the numbers (usually positive or non-negative integers). The most common choices of domain are the set of all vertices (vertex labelings), the edge set alone (edge labelings), or the set of all vertices and edges (total labelings). In many cases, it is interesting to consider the sum of all labels associated with a graph element. It is called the weight of the element: the weight of a vertex or the weight of an edge. In this thesis, we consider a total k-labeling as a labeling of the vertices and edges of graph G with labels from the set {1, 2, . . . , k}. A total k-labeling is defined to be an edge irregular total k-labeling of the graph G if edge-weights are different for all pairs of distinct edges and to be a vertex irregular total k-labeling of G if vertex-weights are different for all pairs of distinct vertices. The minimum value of k for which the graph G has an edge irregular total k- labeling is called the total edge irregularity strength of the graph G, tes(G). Analo- gously, the total vertex irregularity strength of G, tvs(G), is defined as the minimum k for which there exists a vertex irregular total k-labeling of G. In this thesis, we present new results on the total edge irregularity strength and the total vertex irregularity strength.
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المبحث الرابع: شعرها

المبحث الرابع: شعرها

 حصلت الشاعرۃ علی فن الشعر منذ صغرھا، وخاصۃً من أجدادها لأمها، وحسن عسکري لہ الدور الکبیر في تربیۃ بروین شاکر وتھذیبھا من الناحیۃ الأدبیۃ کانت الشاعرۃ تحب ان تقرأ من الشعر أو ما شابہ ذلک من الجرائد والمجلات وکانت تحب ان تسمع القصائد والأغاني في الرادیو وکانت تحب ان تُردد خلف الأغاني والاشعار۔ وعندما وصلت مرحلۃ الثانویۃ تقابلت مع الشاعرۃ المعروفۃ عرفانہ عزیز، فقامت عرفانہ بتربیۃ بروین من الناحیۃ الأدبیۃ والشعریۃ وتھذیب ألفاظھا، ومنحتھا ببعض النصائح والإرشادات التي أفادتھا في مستقبلھا۔ وأول عمل شعري لبروین شاکر کان في 6 - سبتمبر وکانت قصیدتھا الأولیٰ المعنونۃ ب(صبحِ وطن) ۔ (صباح الوطن) وکانت حول 6- سبتمبر، ویقول الدکتور ناظم جعفري عن بروین بأنھا تربت وترعرعت تحت رعایۃ حسن عسکري[1].

 تعرفت الشاعرۃ علی أحمد ندیم قاسمي عن طریق منتج رادیو پاکستان یاور مھدي ویقول أحمد ندیم قاسمي وصلتني رسالۃ من شاعر شاب عبیداﷲ علیم في 1971م وفیہ بعض أشعار لبروین شاکر ویقول ھذا الشاب عن بروین شاکر أنھا شابۃ بارعۃ ولھا قصائد رائعۃ، فیقول أحمد ندیم عندما وصتلني أشعارھا وبعد قراءتھا صححتُ بعض الأخطاء البسیطۃ ثم تم نشرها في 1971م[2]، وساعدھا الأستاذ القاسمي في المھارۃ الفنیۃ والشعریۃ وأعطاھا الکثیر من النصائح والإرشادات المھمۃ التي ساعدتھا في براعتھا وتھذیب لُغۃ الشعر وتطور أسالبیھا الشعریۃ۔

 



[1] الدکتور ناظم جعفري، خوشبو کی ہمسفر، ص78ـ79

[2] أحمد ندیم قاسمي، خوشبو کی ہمسفر، ص17۔

مصادر قانون بطور سند اسلامی اور مغربی اصول قانون كا تحقیقی اور تقابلی جائزہ

Validity of the law depends on its derivation from legitimate sources. The term ‘source’ denotes the norm that validates a law. Western law is based on western legal tradition which is deeply rooted in Roman law and Bible. Statutes is one of the basic source of western law, however, constitution is superior source of western legal system. The sources of Islamic law, unlike to western legal system, are basically divided into primary and secondary sources. This study aims to compare the authority of sources of legal systems, Islam and western, and analyze the objections of orientalists on Islamic law and its sources.

Evaluation of Biochar As a Carrier Material of Antagonistic Rhizobacteria for the Management of Root Pathogenic Fungi of Chickpea

Root diseases either soil or seed borne poses serious implications on seed germination and subsequently plant growth and development. The attack of root diseases in earlier stages of the crop completely destroys the crop resulting in unrecoverable losses. Biological control offers great potential to resist the pathogenic microbes to develop in rhizosphere subsequently reducing the attack. However persistency and sustainable number of biological control agent must be there to fight against the pathogens. In this study we determined the biochar as a carrier of antagonistic rhizobacteria and how much effective biochar is in sustaining the bacterial count in the rhizosphere and also during storage. We further determined the effect of biochar on root nodulating factors that are being altered by biochar alone and also with bacterial inoculant. This study was initiated with survey to determine the prevalence, incidence and severity of major fungal root pathogens infecting chickpea crop in major chickpea growing areas of Punjab i.e. districts Layyah and Bhakkar. Surveys were conducted during year 2013-14 and 2014-15 in the months of November to March. From each district twenty areas and three fields from each area were visited making total 120 fields. Results revealed 100% prevalence of root diseases in every single field and average highest incidence in Layyah was 60.05% and in Bhakkar it was 57.5%. The minimum and maximum incidence in Layyah and Bhakkar was 52%-72%, 48%-76% and 48%-76%, 50%-70% during 2013-14 and 2014-15 respectively. Disease severity ranged from 1-4 in almost every area/field visited. Collected samples retrieved five fungal pathogens and the most frequent was F. oxysporum f. sp. ciceri, with average highest frequency of 88.5% followed by M. phaseolina and Verticillium spp. with 46.5% and 24.5% respectively. Results xxii elucidated that fungal pathogens associated to roots of chickpea pose serious implications on growth and productivity of chickpea. The rhizobacteria i.e. Paenibacillus illinoisensis Rh-31 and Mesorhizobium ciceri antagonistic to fungal pathogens were inoculated to several biochar (Wheat straw (WS), Sheesham wood (SW), Sugarcane bagasse (SB) and Acacia wood (AW)) and their concentration was investigated at regular intervals of 30 days till 180 days. Highest bacterial count (cfu/g) was obtained with SB biochar even after 180 days. It was noticed that biochar held much more bacterial cells as compared to peat and pH of the carrier material seemed to be the key factor in determining the ability of compound to serve as carrier. The impact of carrier materials on disease and plant growth was studied under greenhouse conditions. SB Biochar with rhizobacteria showed significant results in decreasing the disease incidence and increasing plant growth. The effect of biochar on nodulating genes was studied by qPCR and it was noticed that expression of all genes was upregulated in real time PCR and synergistic treatments (Mesorhizobium ciceri + SB biochar) plants showed noteworthy difference from non-inoculated and independently treated with rhizobia and biochar plants. Further testing of biochar under field conditions also showed better growth as compared to non-treated plants. Therefore on the basis of results it is concluded that sugarcane bagasse biochar can be used as a good carrier material of bacterial inoculant for long term storage and also for application in the field.