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Multilevel and Antipodal Labelings for Families of Graphs

Thesis Info

Access Option

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Author

Nazeer, Saima

Program

PhD

Institute

Lahore College for Women University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2016

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/10245/1/Saima%20Nazeer_LGWU_2015.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726732580

Similar


A radio k-labeling c of a graph G is a mapping c : V (G) → Z+ ∪ {0}, such that d(x, y) + |c(x) − c(y)| ≥ k + 1 holds for every two distinct vertices x and y of G, where d(x, y) is the distance between any two vertices x and y of G. The span of a radio k-labeling c is denoted by sp(c) and defined as max{|c(x) − c(y)| : x, y ∈ V (G)}. The radio labeling is a radio klabeling when k = diam(G). In other words, a radio labeling is a one-to-one function c : V (G) → Z+ ∪ {0}, such that |c(x) − c(y)| ≥ diam(G) + 1 − d(x, y) for any pair of vertices x, y in G. The radio number of G denoted by rn(G), is the lowest span taken over all radio labelings of the graph. When k = diam(G) − 1, a radio klabeling is called a radio antipodal labeling. An antipodal labeling for a graph G is a function c : V (G) → {0, 1, 2, ...}, so that d(x, y) + |c(x) − c(y)| ≥ diam(G) for all x, y ∈ G. The radio antipodal number for G denoted by an(G), is the minimum span of an antipodal labeling admitted by G. In this thesis, we investigate the exact value of the radio number and radio antipodal number for different family of graphs. Further more, we also determine the lower bound of the radio number for some cases.
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مرزا نیاز احمد بیگ

مرزا نیاز احمد بیگ
افسوس ہے کہ ۷؍ فروری کو دارالمصنفین کے ایک قدیم رکن مرزا نیاز احمد بیگ بھی رحلت فرماگئے، وہ اعظم گڑھ کے ممتاز اور کامیاب وکیل اور شہر کے عمائد میں تھے، مولانا شبلیؒ اور ان سے نسبت رکھنے والے تمام اداروں سے ان کو بڑا تعلق تھا، شبلیؒ نیشنل پوسٹ گریجویٹ کالج کی مجلس انتظامیہ کے برسوں رکن اور نائب صدر رہے، اب ان کی تمام تر توجہ دارالمصنفین کی طرف مرکوز ہوگئی تھی جس کی مجلس انتظامیہ کے وہ آخر دم تک رکن رہے، دارالمصنفین کے نازک اور بحرانی دور میں ان کے مفید قانونی مشوروں سے اس کو بڑا فائدہ پہنچا، اب وہ اپنے ایک سچے بہی خیرخواہ اور مخلص ہمدرد سے محروم ہوگیا، وہ صوم و صلوٰۃ کے ہمیشہ سے پابند رہے، حج بیت اﷲ سے بھی مشرف ہوئے، آخر عمر میں ان کی دینداری زیادہ بڑھ گئی تھی، علماء و صلحاء سے بھی تعلق رکھتے تھے، حضرت مولانا سید ابوالحسن علی ندوی سے خاص عقیدت تھی، اﷲ تعالےٰ ان کی بشری لغزشوں سے درگذر فرمائے۔ اور انھیں اپنے جوار رحمت میں جگہ دے۔
(ضیاء الدین اصلاحی، فروری ۱۹۸۸ء)

 

سیرت نبویﷺ کے منتخب موضوعات (نبوت، وحی اور معجزات) سے متعلق لزلے ہزلٹن کے آراء کا تحقیقی جائزہ

All mighty Allah (SWT) has created man and made him crown of the nature. For the physical needs of human being, Allah provided him all the needful things. In the same way, Allah (SWT) sent his prophets for the spiritual guidance of mankind. Prophet Muhammad (SAW) was the last messenger of this chain of the prophets. The Prophet hood of Muhammad (SAW) was for the whole of the mankind, for all the people till the day of judgment. Because Allah (SWT) called him the “Seal of the prophets”. And it the gift of Allah to whom he wants to give, he gives. Nobody can achieve this (prophet hood) by its struggle. All the prophets of Allah (SWT) were given instructions through revelation. Some people, who were unable to accept Islam, would make jokes of the teachings of the Prophet (SAW), but he continued his mission and the world witnessed a day when Islam spread all over the world. Some non-Muslims, specially the orientalists, tried to pin point some aspects of the life of the Holy Prophets (in their thinking) but as Allah (SWT) said: And we raised high your name. (Al-Quran: 94: 4) Lesley Hazelton is one of those orientalists. She wrote many books about Islam. Her famous book is “The First Muslim-The Story of Muhammad” in which she has targeted many aspects of the Holy Prophet Muhammad (SAW). This article discusses her views about the Prophet hood, Revelation and the Miracles of the Holy Prophet (SAW). It is out of the human mind to understand the reality of the Prophet hood and revelation. It is the gift of Allah to whom he wants, He gives prophet hood. The family, fame, wealth and physical strength cannot be the reasons for the achievement of Prophet hood. The human mind cannot grab the reality of the revelation. 

پاکستان میں دفتری اردو کا تجزیاتی مطالعہ

The status of Urdu as an official language has a history that goes back to the first quarter of 19th century. At that time Urdu enjoyed the status of official language in some parts of the sub-continent. The court of Maharaja Ranjit Singh was linked with Urdu in quite a few ways. The Maharaja and his ancestors patronized the Urdu writers and poets as their courtiers. Even if the official language of the Maharaja's court was Persian the ruler himself employed Panjabi for daily administrative matters. But he used Urdu language while conversing with the visitors, the tourists, the diplomats from friendly countries and also with the British officers. From this it is easy to surmise that even if Urdu was not the official languages of his court it was still the language of everyday use for courtly and informal occasions. In the thesis under review an attempt has been made to study Urdu in the ways it has been used as an official language, in various times and eras. When the Quaid-e Azam declared Urdu as an official language of Pakistan such a study becomes imperative. There is a need to study Urdu as an official language in its various styles and this thesis makes a humble attempt to fulfill the need. The aim is to analyse Urdu as official language, particularly in view of the Quaid's declaration, and to study its significance as such. During the writing of the thesis the research matter has been analysed in a historical context. Every effort has been made to ensure the authenticity of the sources and these encompass the unpublished as well as the published material from antiquity to the modern times. In this way a period of two centuries have been covered to reflect to real purpose of the thesis.