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Face Labelings of Graphs Embedded on the Surface of a Sphere or a Torus

Thesis Info

Access Option

External Link

Author

Numan, Muhammad

Supervisor

Martin Baca

Program

PhD

Institute

Government College University

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2012

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/2285/1/2516S.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726193890

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The thesis deals with the problem of labeling the vertices, edges and faces of a plane graphs by the consecutive integers in such a way that the label of a face and the labels of the vertices and edges surrounding that face all together add up to a weight of that face. If these face weights form an arithmetic progression with common difference d then the labeling is called d-antimagic. Such a labeling is called super if the smallest possible labels appear on the vertices. The thesis examines the existence of such labelings for toroidal fullerenes, gener- alized prism and disjoint union of generalized prisms. The toroidal fullerene is a 2-colorable cubic graph, there exist a 1-factor (perfect matching) and a 2-factor (a collection of n cycles on 2m vertices each). First we label the vertices of toroidal fullerene and then we label the edges of a 1-factor by consecutive integers and then in successive steps we label the edges of 2m-cycles (respectively 2n-cycles) in a 2-factor by consecutive integers. This technique allows us to construct super d-antimagic labelings of type (1, 1, 1) of toroidal fullerenes for several values of d. We consider the generalized prism as a collection of two classes of cycles: the main cycles and the middle cycles. To label the main cycles and the middle cycles we use the super (a, d)-edge-antimagic total and (a, d)-edge-antimagic total labelings and combine these labelings to a resulting super d-antimagic labeling of type (1, 1, 1). The disjoint union of generalized prism can be considered as a collection of disjoint union of main cycles and disjoint union of middle cycles. To label the disjoint union of main and middle cycles we again use edge-antimagic total labelings and super edge- antimagic total labelings. Combining these labelings we obtain a resulting super d-antimagic labeling of type (1, 1, 1) for a given diference d.
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نقشِ پائے شہِ ابرار قمر کی رونق

نقشِ پائے شہِ ابرارؐ ؛ قمر کی رونق
کہکشائوں میں ڈھلی ، راہ گزر کی رونق

یہی دیواروں کی زینت ، یہی در کی رونق
ذکرِ سرکارِؐ دو عالم سے ہے گھر کی رونق

نقشِ نعلین نبیؐ سے ہے مدینے کی چمک
اسی دستار سے ہے میرے بھی سر کی رونق

رحمتِ کون و مکاںؐ جانِ دو عالم ٹھہری
یادِ محبوبِ خداؐ دل کے نگر کی رونق

آپؐ کے دم سے ہے اے صاحبِ لولاک لماؐ!
بحر و بر ، حجر و شجر ، شام و سحر کی رونق

لفظِ کُن ، روزِ ازل ، لوح و قلم ، عرشِ عُلی
آپؐ مخلوقِ خدا ، جنّ و بشر کی رونق

عدسئہ آنکھ میں ہے نورِ مُنوّر اُنؐ کا
نورِ محبوبِ خداؐ ، حسنِ نظر کی رونق

روزنِ چشمِ عقیدت سے حقیقت دیکھو
ذاتِ محبوبؐ سے اللہ کے گھر کی رونق

رونقِ چہرئہ اُمّت ہے شفاعت کے سبب
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