Resultados espectrais relacionados com a estrutura dos grafos

Nesta tese são estabelecidas novas propriedades espectrais de grafos com estruturas específicas, como sejam os grafos separados em cliques e independentes e grafos duplamente separados em independentes, ou ainda grafos com conjuntos (κ,τ)-regulares. Alguns invariantes dos grafos separados em cliques e independentes são estudados, tendo como objectivo limitar o maior valor próprio do espectro Laplaciano sem sinal. A técnica do valor próprio é aplicada para obter alguns majorantes e minorantes do índice do espectro Laplaciano sem sinal dos grafos separados em cliques e independentes bem como sobre o índice dos grafos duplamente separados em independentes. São fornecidos alguns resultados computacionais de modo a obter uma melhor percepção da qualidade desses mesmos extremos. Estudamos igualmente os grafos com um conjunto (κ,τ)-regular que induz uma estrela complementar para um valor próprio não-principal $. Além disso, é mostrado que $=κ-τ. Usando uma abordagem baseada nos grafos estrela complementares construímos, em alguns casos, os respectivos grafos maximais. Uma caracterização dos grafos separados em cliques e independentes que envolve o índice e as entradas do vector principal é apresentada tal como um majorante do número da estabilidade dum grafo conexo.

Polinómios de Appell multidimensionais e sua representação matricial

Nesta dissertação é apresentada uma abordagem a polinómios de Appell multidimensionais dando-se especial relevância à estrutura da sua função geradora. Esta estrutura, conjugada com uma escolha adequada de ordenação dos monómios que figuram nos polinómios, confere um carácter unificador à abordagem e possibilita uma representação matricial de polinómios de Appell por meio de matrizes particionadas em blocos. Tais matrizes são construídas a partir de uma matriz de estrutura simples, designada matriz de criação, subdiagonal e cujas entradas não nulas são os sucessivos números naturais. A exponencial desta matriz é a conhecida matriz de Pascal, triangular inferior, onde figuram os números binomiais que fazem parte integrante dos coeficientes dos polinómios de Appell. Finalmente, aplica-se a abordagem apresentada a polinómios de Appell definidos no contexto da Análise de Clifford.

Spectral characterization of families of split graphs

An upper bound for the sum of the squares of the entries of the principal eigenvector corresponding to a vertex subset inducing a k-regular subgraph is introduced and applied to the determination of an upper bound on the order of such induced subgraphs. Furthermore, for some connected graphs we establish a lower bound for the sum of squares of the entries of the principal eigenvector corresponding to the vertices of an independent set. Moreover, a spectral characterization of families of split graphs, involving its index and the entries of the principal eigenvector corresponding to the vertices of the maximum independent set is given. In particular, the complete split graph case is highlighted.

Necessary and sufficient conditions for a Hamiltonian graph

A graph is singular if the zero eigenvalue is in the spectrum of its 0-1 adjacency matrix A. If an eigenvector belonging to the zero eigenspace of A has no zero entries, then the singular graph is said to be a core graph. A ( k,t)-regular set is a subset of the vertices inducing a k -regular subgraph such that every vertex not in the subset has t neighbours in it. We consider the case when k=t which relates to the eigenvalue zero under certain conditions. We show that if a regular graph has a ( k,k )-regular set, then it is a core graph. By considering the walk matrix we develop an algorithm to extract ( k,k )-regular sets and formulate a necessary and sufficient condition for a graph to be Hamiltonian.

Numerical and combinatorial applications of generalized Appell polynomials

This thesis studies properties and applications of different generalized Appell polynomials in the framework of Clifford analysis. As an example of 3D-quasi-conformal mappings realized by generalized Appell polynomials, an analogue of the complex Joukowski transformation of order two is introduced. The consideration of a Pascal n-simplex with hypercomplex entries allows stressing the combinatorial relevance of hypercomplex Appell polynomials. The concept of totally regular variables and its relation to generalized Appell polynomials leads to the construction of new bases for the space of homogeneous holomorphic polynomials whose elements are all isomorphic to the integer powers of the complex variable. For this reason, such polynomials are called pseudo-complex powers (PCP). Different variants of them are subject of a detailed investigation. Special attention is paid to the numerical aspects of PCP. An efficient algorithm based on complex arithmetic is proposed for their implementation. In this context a brief survey on numerical methods for inverting Vandermonde matrices is presented and a modified algorithm is proposed which illustrates advantages of a special type of PCP. Finally, combinatorial applications of generalized Appell polynomials are emphasized. The explicit expression of the coefficients of a particular type of Appell polynomials and their relation to a Pascal simplex with hypercomplex entries are derived. The comparison of two types of 3D Appell polynomials leads to the detection of new trigonometric summation formulas and combinatorial identities of Riordan-Sofo type characterized by their expression in terms of central binomial coefficients.

Superregular matrices and applications to convolutional codes

The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matrix is superregular if all of its minors that are not trivially zero are nonzero. Given a a×b, a ≥ b, superregular matrix over a field, we show that if all of its rows are nonzero then any linear combination of its columns, with nonzero coefficients, has at least a−b + 1 nonzero entries. Secondly, we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code, namely, the rate and the Forney indices. These results answer some open questions on distances and constructions of convolutional codes posted in the literature.