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Total and Entire Labeling of Graphs

Thesis Info

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Author

Naseem, Maria.

Program

PhD

Institute

Government College University Lahore

City

Lahore

Province

Punjab

Country

Pakistan

Thesis Completing Year

2017

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/9456/1/Maria_Naseem_Maths_2017_GCU_Lahore_22.11.2017.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676727618617

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\A plane graph is a particular drawing of a planar graph on the Euclidean plane. Let G(V;E; F) be a plane graph with vertex set V , edge set E and face set F. A proper entire t-colouring of a plane graph is a mapping : V (G) [ E(G) [ F(G) ?! f1; 2; : : : ; tg such that any two adjacent or incident elements in the set V (G)[E(G)[F(G) receive distinct colours. The entire chromatic number, denoted byvef (G), of a plane graph G is the smallest integer t such that G has a proper entire t-colouring. The proper entire t-colouring of a plane graph have been studied extensively in the literature. There are several modi cation on entire t-colouring. We focus on a face irregular entire k-labeling of a 2-connected plane graph as a labeling of vertices, edges and faces of G with labels from the set f1; 2; : : : ; kg in such a way that for any two di erent faces their weights are distinct. The weight of a face under a k-labeling is the sum of labels carried by that face and all the edges and vertices incident with the face. The minimum k for which a plane graph G has a face irregular entire k-labeling is called the entire face irregularity strength". \Another variation to entire t-colouring is a d-antimagic labeling as entire labeling of a plane graph with the property that for every positive integer s, the weights of s-sided faces form an arithmetic sequence with a common di erence d. In the thesis, we estimate the bounds of the entire face irregularity strength for disjoint union of multiple copies of a plane graph and prove the sharpness of the lower bound in two cases. Also we study the existence of d-antimagic labelings for vi the Klein-bottle fullerene that is for anite trivalent graph embedded on the Kleinbottle with each face is a hexagon. In last chapter we investigate the 3-total edge product cordial labeling of hexagonal grid (honeycomb) that is the planar graph with m rows and n columns of hexagons".
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ہماری دعا میں اثر ہی کہاں تھا

ہماری دُعا میں اثر ہی کہاں تھا
بھلا بے وفا وہ بشر ہی کہاں تھا

میں گھر کیوں نہیں جاتا سب پوچھتے ہیں
کہوں اب میں کیا میرا گھر ہی کہاں تھا

تجھے مانگنے کے سوا دشمنِ جاں
گرا اپنا سجدے میں سر ہی کہاں تھا

جو آباد تھا کس قدر اُس کے دم سے
بغیر اُس کے ویسا نگر ہی کہاں تھا

زمانے کی حدت سے مجھ کو بچاتا
رہِ عشق میں وہ شجر ہی کہاں تھا

پاکستان میں جبری شادیوں كے اسباب و محرکات

Before the advent of Islam, a woman was considered as property, with no rights at all. They did not have the right to choose a husband. Islam liberated women and gave them the basic rights. Unfortunately, in Pakistan many parents and legal guardians misinterpret and misuse the guardianship of their children, especially with reference to marriage. They deny some religious rights to their children because of local traditions, customs and taboos. Forcing children to marry against their wishes, does not conform to the Islamic code. Forced marriages are considered illegle in Shariah. Even in the case of marriage of a minor arranged, by a father or a grandfather, the girl has the right to cancal it on reaching Puberty. According to Shariah the boy and the girls has the right to turn to a court for a settlement of the issue.

Conventional and Microwave-Assisted Synthesis of Trisubstituted-1, 2, 4-Triazoles and Their Structure-Activity Relationship Studies

The chemistry of life has been extensively and effectively elaborated by organic chemists. No doubt, all the fields of science are struggling for the benefit of human beings and the chemists are well known in this regard. The organic or medicinal chemists are the main worker of improving the health. A number of compounds have been synthesized in search of new drug candidates by the orgnic chemists and also they have been successful up to much extent in this regard. The importance of this process can be realized from the increasing inactivity of the running drugs against different diseases. Hence the synthesis of new molecules in search of new drug candidates against different diseases is an ever green process. This was the motivating aspect of the literature survey of synthetic chemistry which encouraged us to design new molecules and evaluate their biological potential. The bioactivity potential of some heterocyclic moieties (as discussed in introduction and review of literature) prompted has to design such type of molecules which bear more than one heterocyclic moieties. These considered heterocyclic moieties included piperidine and 1,2,4-triazole. The aim of submerging different heterocyclic functionalities into one core was to boost up their bioactivity potential. Furthermore, the variation in some part of final molecules was also processed in order to acquire new potent drug candidates. The pharmacological evaluation included enzyme inhibition, antioxidant activity and bovine serum albumin (BSA) binding analysis. The enzyme inhibition results were further substantiated through molecular docking analysis. The presented research work has been distributed into eight solid schemes for the synthesis of ninety six compounds. Ethyl isonipecotate (2) was treated with 4methoxybenzene sulfonyl chloride (1) in 5% sodium carbonate at pH of 9-10 to get ethyl-1-[(4-methoxyphenyl)sulfonyl]piperidine-4-carboxylate (3). Compound 3 and hydrazine monohydrate were refluxed in methanol to acquire 1-[(4methoxyphenyl)sulfonyl]piperidine-4-carbohydrazide (4). Compound 4 was refluxed with phenyl isothiocyanate in methanol to acquire an intermediate compound (2-({1[(4-methoxyphenyl)sulfonyl]-4-piperidinyl}carbonyl)-N-phenyl-1-hydrazinecarbothio amide) which was refluxed in basic medium to get 5-{1-[(4-methoxyphenyl)sulfonyl]4-piperidinyl}-4-phenyl-4H-1,2,4-triazole-3-thiol (5). Compound 5 was stirred with xiii different aralkyl halides (6a-j) in the presence of NaH and DMF using conventional and microwave assisted methods. 3-Aralkylthio-5-{1-[(4-methoxyphenyl)sulfonyl]-4piperidinyl}-4-phenyl-4H-1,2,4-triazole (7a-j) were obtained through filtration from aqueous medium. The compound 5 was treated with equimolar N-substituted-2bromoacetamides (10a-t) to acquire N-alkyl/aralkyl/aryl/phenyl-2-[(5-{1-[(4methoxyphenyl) sulfonyl]-4-piperidinyl}-4-phenyl-4H-1,2,4-triazol-3-yl)sulfanyl] acetamide (11a-t). The electrophiles, 10a-t, were synthesized by the reaction of alkyl/aralkyl/aryl/phenyl amines (8a-t) and bromoacetyl bromide (9) in 5% sodium carbonate solution. The compound 5 was treated with equimolar N-substituted-2bromopropanamides (13a-r) to acquire N-alkyl/aralkyl/aryl/phenyl-2-[(5-{1-[(4methoxyphenyl) sulfonyl]-4-piperidinyl}-4-phenyl-4H-1,2,4-triazol-3-yl)sulfanyl] propanamide (14a-r). The electrophiles, 13a-r, were synthesized by the reaction of alkyl/aralkyl/aryl/phenyl amines (8a-i, k, m-t) and 2-bromopropionyl bromide (12) in 5% sodium carbonate solution. Compound 4 was refluxed with methyl isothiocyanate in methanol to acquire an intermediate compound (2-({1-[(4-methoxyphenyl) sulfonyl]-4-piperidinyl}carbonyl)-N-methyl-1-hydrazinecarbothioamide) which was refluxed in basic medium to get 5-{1-[(4-methoxyphenyl)sulfonyl]-4-piperidinyl}-4methyl-4H-1,2,4-triazole-3-thiol (15). Compound 15 was stirred with different aralkyl halides (6a-j) in the presence of NaH and DMF using conventional and microwave assisted methods. 3-Aralkylthio-5-{1-[(4-methoxyphenyl)sulfonyl]-4-piperidinyl}-4methyl-4H-1,2,4-triazole (16a-j) were obtained through filtration from aqueous medium. The compound 15 was treated with equimolar N-substituted-2bromoacetamides (10a-t) to acquire N-alkyl/aralkyl/aryl/phenyl-2-[(5-{1-[(4methoxyphenyl) sulfonyl]-4-piperidinyl}-4-methyl-4H-1,2,4-triazol-3-yl)sulfanyl] acetamide (17a-t). The compound 15 was treated with equimolar N-substituted-2bromopropanamides (13a-r) to acquire N-alkyl/aralkyl/aryl/phenyl-2-[(5-{1-[(4methoxyphenyl) sulfonyl]-4-piperidinyl}-4-methyl-4H-1,2,4-triazol-3-yl)sulfanyl] propanamide (18a-r). The synthesized compounds were initially verified through TLC and stored for further analysis. The synthesized compounds were spectroscopically characterized by using IR, 1H-NMR, 13C-NMR, HMQC, HMBC, COSY, NOESY and EIMS spectral information to justify the available main functional groups, hydrogen atoms, carbon atoms and the fragmentation pattern of the structures of synthesized compounds. xiv The synthesized compounds were screened for enzyme inhibition activity against six different enzymes and also for antioxidant activity. The different six enzymes included acetyl cholinesterase (AChE), butyryl cholinesterase (BChE), αglucosidase, urease, lipoxygenase and carbonic anhydrase II enzyme. Almost all the compounds were found to be excellent active agents against these enzymes. Antioxidant activityof all the synthesized molecules wasalso tested in search of some unique drug candidates. The chemistry of active sites and different functionalities responsible for the best pharmacological potential of all the synthesized compounds was verified through docking studies. In addition to it, the evaluation of protein drug interaction assisted us in understanding the various binding sites and binding constant to justify the stay of the drugs in the body, their circulation, metabolism, elimination and pharmacodynamics. The sketched compounds in the eight schemes were synthesized efficiently with high yield and purity through environment friendly protocol with minimum cost and time. The time of synthesis and the yield were compared for two modes of synthetic methods including conventional and microwave assisted ones. The following synthetic as well as biological screening studies resulted into the identification of a numberof compoundsbeing active against the considered enzymes. These enzymes are responsible for different kind of diseases and so the bioactive potent compounds may be considered as new drug candidates for the concerned diseases.