This thesis explores 2-generated and cubic Cayley graphs. All 2-generated Cayley graphs with generators from Sn where n < 9, were generated. Further, 3-generated cubic Cayley graphs, where n < 7, were also generated. Among these, the cubic Cayley graphs with up to 40320 vertices were tested for various properties including Hamiltonicity and diameter. These results are available on the internet in easy to read tables. The motivation for the testing of Cayley graphs for Hamiltonicity was the conjecture that states that every connected Cayley graph is Hamiltonian. New enumeration results are presented for various classes of 2-generated Cayley graphs. Previously known enumeration results are presented for cubic Cayley graphs.Finally, isomorphism and color isomorphism of 2-generated and cubic Cay­ley graphs is explored. Numerous new results are presented. All algorithms used in this thesis are explained in full.

This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital operator algebras A and B. Because of the different properties enjoyed by the two tensor products in connection to dilation theory, it is of interest to know when they coincide (completely isometrically). Motivated by earlier work due to Paulsen and Power, we provide conditions relating an operator algebra B and another family {C_i}_i of operator algebras under which, for any operator algebra A, the equality A ⊗_max B = A ⊗_min B either implies, or is implied by, the equalities A ⊗_max C_i = A ⊗_min C_i for every i. These results can be applied to the setting of a discrete group G pre-ordered by a subsemigroup G⁺, where B ⊆ C*_r(G) is the subalgebra of the reduced group C*-algebra of G generated by G⁺, and C_i = A(Q_i) are digraph algebras defined by considering certain pre-ordered subsets Q_i of G. The 16-dimensional algebra A₄ of real sedenions is obtained by applying the Cayley-Dickson doubling process to the real division algebra of octonions. The classification of subalgebras of A₄ up to conjugacy (i.e. by the action of the automorphism group of A₄) was completed in a previous investigation, except for the collection of those subalgebras which are isomorphic to the quaternions. We present a classification of quaternion subalgebras up to conjugacy.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [51]-52).

Thesis (Ph.D.)--Mississippi State University. Department of Mathematics and Statistics. / Title from title screen. Includes bibliographical references.

Um pencil de Halphen é uma família a um parâmetro de curvas sêxticas planas com nove pontos duplos pré-fixados. Estes nove pontos não podem ser escolhidos ao acaso: fixados oito em posição geral, o nono deve pertencer à curva dianodal de Cayley. Neste trabalho abordamos diferentes métodos de construção da curva dianodal. Estudamos também a superfície dianodal, lugar geométrico de um oitavo ponto duplo isolado de superfícies quárticas de CP³. Estes assuntos são relacionados com as involuçães de Bertini e Kantor. / A Halphen peneil is a one parameter family of plane sextic curves with nine fixed double points. These nine points can't be chosen arbitrarily: fixed eight in general position, the ninth must lie on Cayley's dianodal curve. In this work we approach different methods to obtain the dianodal curve. We aIso study the dianodal surface, the locus of an eighth isolated triple point of quartic surfaces in CP³. These subjects are related with Bertini and Kantor involutions.

Um pencil de Halphen é uma família a um parâmetro de curvas sêxticas planas com nove pontos duplos pré-fixados. Estes nove pontos não podem ser escolhidos ao acaso: fixados oito em posição geral, o nono deve pertencer à curva dianodal de Cayley. Neste trabalho abordamos diferentes métodos de construção da curva dianodal. Estudamos também a superfície dianodal, lugar geométrico de um oitavo ponto duplo isolado de superfícies quárticas de CP³. Estes assuntos são relacionados com as involuçães de Bertini e Kantor. / A Halphen peneil is a one parameter family of plane sextic curves with nine fixed double points. These nine points can't be chosen arbitrarily: fixed eight in general position, the ninth must lie on Cayley's dianodal curve. In this work we approach different methods to obtain the dianodal curve. We aIso study the dianodal surface, the locus of an eighth isolated triple point of quartic surfaces in CP³. These subjects are related with Bertini and Kantor involutions.

Made available in DSpace on 2014-06-11T19:24:51Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-10-15Bitstream added on 2014-06-13T19:11:41Z : No. of bitstreams: 1 godoy_kv_me_rcla.pdf: 3111076 bytes, checksum: 9f80d8c7c4d74c8d285fc6f1c78118a7 (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A finalidade desta dissertação é estudar o reconhecimento histórico atribuído ao matemático Arthur Cayley na História da Matemática. Iniciamos este trabalho relatando sua trajetória escolar no King´s College de Londres e Trinity College, em Cambridge. Em 1852, Cayley foi eleito membro da Royal Society of London. Portanto, realizamos um estudo em relação ao seu processo de admissão e do estado da Royal Society of London nesse período. É possível encontrar o nome de Arthur Cayley nos principais mecanismos de reconhecimento matemático da época, portanto, verificamos esse reconhecimento por meio dos pareceres relativos às suas publicações abordando a Teoria dos Invariantes na Philosophical Transactions of Royal Society of London. Fora da Inglaterra, o reconhecimento matemático de Cayley é constatado pelas suas publicações no Crelle (Alemanha), Liouville e Comptes rendus (França). Devido ao seu prestígio, Cayley foi convidado para publicar em revistas matemáticas da Itália e Estados Unidos. O reconhecimento acadêmico de Arthur Cayley é decretado em 1863, quando foi eleito à primeira Cadeira de Professor Sadleirian de Matemática Pura da Universidade de Cambridge. Ao final de tratar essas questões, finalizamos com uma discussão sobre o processo de reconhecimento histórico na Matemática, em especial, o caso de Arthur Cayley / The aim of this dissertation is to study the historical recognition awarded to the mathematician Arthur Cayley in the History of Mathematics. We started this work reporting their trajectory at King's College London and at Trinity College, Cambridge. In 1852, Cayley was elected Fellow of the Royal Society of London. Therefore, we conducted a study regarding their admissions process and the state of the Royal Society of London in this period. It´s possible find the name of Arthur Cayley in the main recognition mechanisms mathematician of the time, therefore, we see this recognition by means of referees concerning about their publications addressing the Invariant Theory in Philosophical Transactions of the Royal Society of London. Outside England, the recognition mathematician from Cayley is remarkable for its publications in Crelle (Germany), Liouville and Comptes rendus (France). Because of its prestige, Cayley was invited to publish in mathematical journals from Italy and the United States. The academic recognition of Arthur Cayley is decreed in 1863, when he was elected to the first Sadleirian Chair Professor of Pure Mathematics at Cambridge University. At the end of dealing with these issues, we concluded with a discussion of the process of historical recognition in mathematics, in particular the case of Arthur Cayley / FAPESP: 11/05133-3

Um pencil de Halphen é uma família a um parâmetro de curvas sêxticas planas com nove pontos duplos pré-fixados. Estes nove pontos não podem ser escolhidos ao acaso: fixados oito em posição geral, o nono deve pertencer à curva dianodal de Cayley. Neste trabalho abordamos diferentes métodos de construção da curva dianodal. Estudamos também a superfície dianodal, lugar geométrico de um oitavo ponto duplo isolado de superfícies quárticas de CP³. Estes assuntos são relacionados com as involuçães de Bertini e Kantor. / A Halphen peneil is a one parameter family of plane sextic curves with nine fixed double points. These nine points can't be chosen arbitrarily: fixed eight in general position, the ninth must lie on Cayley's dianodal curve. In this work we approach different methods to obtain the dianodal curve. We aIso study the dianodal surface, the locus of an eighth isolated triple point of quartic surfaces in CP³. These subjects are related with Bertini and Kantor involutions.

Masters of Science / Codes Related to and Derived from Hamming Graphs T.R Muthivhi M.Sc thesis, Department of Mathematics, University of Western Cape For integers n; k 1; and k n; the graph 􀀀k n has vertices the 2n vectors of Fn2 and adjacency de ned by two vectors being adjacent if they di er in k coordinate positions. In particular, 􀀀1 n is the classical n-cube, usually denoted by H1(n; 2): This study examines the codes (both binary and p-ary for p an odd prime) of the row span of adjacency and incidence matrices of these graphs. We rst examine codes of the adjacency matrices of the n-cube. These have been considered in [14]. We then consider codes generated by both incidence and adjacency matrices of the Hamming graphs H1(n; 3) [12]. We will also consider codes of the line graphs of the n-cube as in [13]. Further, the automorphism groups of the codes, designs and graphs will be examined, highlighting where there is an interplay. Where possible, suitable permutation decoding sets will be given.

We investigate power graphs of quasigroups. The power graph of a quasigroup takes the elements of the quasigroup as its vertices, and there is an edge from one element to a second distinct element when the second is a left power of the first. We first compute the power graphs of small quasigroups (up to four elements). Next we describe quasigroups whose power graphs are directed paths, directed cycles, in-stars, out-stars, and empty. We do so by specifying partial Cayley tables, which cannot always be completed in small examples. We then consider sinks in the power graph of a quasigroup, as subquasigroups give rise to sinks. We show that certain structures cannot occur as sinks in the power graph of a quasigroup. More generally, we show that certain highly connected substructures must have edges leading out of the substructure. We briefly comment on power graphs of Bol loops.