The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix. The concept is related to the energy of a class of molecules in chemistry and was first brought to mathematics by Gutman in 1978 ([8]). In this thesis, we do a comprehensive study on the energy of graphs and digraphs. In Chapter 3, we review some existing upper and lower bounds for the energy of a graph. We come up with some new results in this chapter. A graph with n vertices is hyper-energetic if its energy is greater than 2n−2. Some classes of graphs are proved to be hyper-energetic. We find a new class of hyper-energetic graphs which is introduced and proved to be hyper-energetic in Section 3.3. The energy of a digraph is the sum of the absolute values of the real part of the eigenvalues of its adjacency matrix. In Chapter 4, we study the energy of digraphs in a way that Pe˜na and Rada in [19] have defined. Some known upper and lower bounds for the energy of digraphs are reviewed. In Section 4.5, we bring examples of some classes of digraphs in which we find their energy. Keywords. Energy of a graph, hyper-energetic graph, energy of a digraph. / vii, 80 leaves ; 29 cm

No description available.

Thesis (Ph.D.) -- McMaster University, 1998. / Includes bibliographical references (leaves 149-166). Also available via World Wide Web.

Thesis (Ph.D.) -- University of Newcastle, 2003. / School of Mathematical and Physical Sciences. Includes bibliographical references (p. 161-162). "Also available online".

Thesis (Ph.D. in Information Sciences)--Naval Postgraduate School, September 2007. / Dissertation Advisor(s): Nissen, Mark ; Arquilla, John "September 2007." Includes bibliographical references (p. 239-277). Also available via the Internet.

Diss. (sammanfattning) Uppsala : Univ., 2002. / Härtill 3 uppsatser.

Thesis (M.Sc.) - Carleton University, 2005. / Includes bibliographical references (p. 61-63). Also available in electronic format on the Internet.

Thesis--PlanB (M.S.)--University of Wisconsin--Stout, 1999. / Field project. Includes bibliographical references.

Thesis (M.S.)--Rochester Institute of Technology, 1994. / Typescript. Includes bibliographical references (leaves 77-80).

Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: Galactooligosaccharides are produced by the transglycosylation activity of β-galactosidases (β-gal, EC 3.2.1.23) when utilising lactose as a substrate. They have emerged as important constituents used in the food and pharmaceutical industries owing to their prebiotic properties. Although transglycosylation was discovered in 1951 (Wallenfels 1951), and a number of β-gals have had their transglycosylation activity characterised, the activities of these enzymes are not optimal for industrial use. Their tendency to favour the hydrolytic reaction over the transglycosylation reaction, coupled with the production of shorter chain oligosaccharides has driven scientists to investigate altering protein structure both to increase chain lengths and the amount of oligosaccharide produced at lower substrate concentrations. In an attempt to alter the amount of oligosaccharide produced by a metagenomically derived β-gal belonging to the glycosyl hydrolase 2 family, random and site-directed mutagenesis were used. A randomly mutagenised library was screened on SOB agar plates containing 5% (w/v) lactose which should select for clones that synthesise oligosaccharides at relatively low concentrations. No such activity was detected. Site-directed mutagenesis was also utilised to alter protein structure. It was confirmed that the β-gal utilised in this study belonged to the glycosyl hydrolase 2 family through mutation of the predicted catalytic acid/base glutamic acid to a non-catalytic residue, thus removing activity. Another mutation was utilised to investigate if it was possible to increase the degree of polymerisation of oligosaccharides produced by the β-gal. This mutation was successful in increasing the degree of polymerisation. Biochemical characterisation of the β-gal revealed that it exhibited optimal activity at pH 8.0, with a temperature optimum of 30°C. The β-gal exhibited a Km and Vmax of 54.23 mM and 2.26 μmol/minute-1/mg protein-1 respectively, similar to kinetic parameters that have been determined for a number of previously characterised enzymes. / AFRIKAANSE OPSOMMING: Galaktooligosakkariede word geproduseer deur die transglikosileering aktiwiteit van β-galaktosidase (β-gal, EG 3.2.1.23) wanneer hulle laktose as 'n substraat gebruik. Hierdie oligosakkariede het na vore gekom as 'n belangrike bestandeel vir gebruik in die voedsel en farmaseutiese bedryf as gevolg van hulle prebiotiese eienskappe. Alhoewel transglycosylation al in 1951 ontdek is (Wallenfels 1951) en 'n aantal β-gals se transglycosylation aktiwiteit gekenmerk is, is hierdie ensieme nie ideaal vir industriële toepassings nie. Die geneigdheid om die hidrolitiese reaksie oor die transglycosylation reaksie bevoordeel, tesame met die produksie van korter oligosakkariede het wetenskaplikes ondersoek genoop om die proteïenstruktuur te verander om ketting-lengte en die kwantiteit van oligosakkaried geproduseer teen laer substraat konsentrasies te verhoog. In 'n poging om die opbrengs van die oligosakkaried wat deur 'n metagenomiese β-gal wat aan die glycosyl hidrolase 2 familie behoort te verander, is lukraak en terrein gerigte-mutagenese gebruik. Die mutagenese biblioteek is op SOB agarplate met 5% (w/v) lactose gekeur, om klone wat die fenotipe wat verband hou met die produksie oligosakkaried teen relatiewe lae konsentrasies te selekteer. Geen aktiwiteit is opgemerk nie. Terrein gerigte-mutagenese is ook gebruik om die proteïenstruktuur te verander. Deur ‘n bioinformatiese voorspelling, is dit bevestig dat die β-gal wat in hiedie studie gebruik word tot die glycosyl hidrolase 2 familie behoort. Dit is gedoen deur mutasie van die voorspelde katalitiese suur/basis glutamiensuur na 'n nie-katalitiese oorskot, dus die verwydering van aktiwiteit. Nog ‘n mutasie is gebruik om te ondersoek of dit moontlik was om die ketting-lengte van die oligosakkaried wat deur die β-gal geproduseer is te verhoog. Die mutasie was suksesvol in die verhoging van die oligosakkaried wat geproduseer was. Biochemiese karakterisering van die β-gal het getoon dat hierdie β-gal optimale aktiwiteit het by pH 8.0, met 'n optimum temperatuur van 30°C. Die β-gal het 'n Km en Vmax van 54.23 mM en 2.26 μmol/minute-1/mg proteïen-1 onderskeidelik, soortgelyk aan kinetiese parameters wat bepaal word vir ensieme wat voorheen gekenmerk is.