Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

Synthesis of structures is a very difficult task even with only a small number of components that form a system; yet it is the catalyst of innovation. Molecular structures and nanostructures typically have a large number of similar components but different connections, which manifests a more challenging task for their synthesis. <p> This thesis presents a novel method and its related algorithms and computer programs for the synthesis of structures. This novel method is based on several concepts: (1) the structure is represented by a graph and further by the adjacency matrix; and (2) instead of only exploiting the eigenvalue of the adjacency matrix, both the eigenvalue and the eigenvector are exploited; specifically the components of the eigenvector have been found very useful in algorithm development. This novel method is called the Eigensystem method.<p> The complexity of the Eigensystem method is equal to that of the famous program called Nauty in the combinatorial world. However, the Eigensystem method can work for the weighted and both directed and undirected graph, while the Nauty program can only work for the non-weighted and both directed and undirected graph. The cause for this is the different philosophies underlying these two methods. The Nauty program is based on the recursive component decomposition strategy, which could involve some unmanageable complexities when dealing with the weighted graph, albeit no such an attempt has been reported in the literature. It is noted that in practical applications of structure synthesis, weighted graphs are more useful than non-weighted graphs for representing physical systems. <p> Pivoted at the Eigensystem method, this thesis presents the algorithms and computer programs for the three fundamental problems in structure synthesis, namely the isomorphism/automorphism, the unique labeling, and the enumeration of the structures or graphs.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / As assinaturas digitais sÃo de fundamental importÃncia para as comunicaÃÃes eletrÃnicas no mundo todo por garantirem a integridade e autenticidade da informaÃÃo. Com os avanÃos da ciÃncia nas Ãreas da mecÃnica quÃntica e a introduÃÃo destes novos conceitos nas telecomunicaÃÃes, a seguranÃa da informaÃÃo tambÃm precisou evoluir e cada vez mais se tem buscado novos sistemas de seguranÃa que forneÃam maior integridade e autenticidade que os sistemas clÃssicos. Dessa forma o objetivo deste trabalho à utilizar as propriedades do problema QSCDff , para a criaÃÃo de um protocolo de assinatura quÃntica de mensagens. O problema QSCD ff possui propriedades matemÃticas e computacionais para garantir a integridade e autenticidade das assinaturas geradas. O protocolo proposto faz uso de chaves descritas na forma de estados quÃnticos construÃdos a partir de permutaÃÃes de um grupo simÃtrico e de uma funÃÃo de hash para a compressÃo da mensagem original. Como entrada o protocolo recebe a mensagem clÃssica e uma chave privada. Para a geraÃÃo do estado quÃntico da assinatura utiliza-se uma permutaÃÃo como chave privada e o hash da mensagem. Gerar tal assinatura sem ter uma chave privada consiste em resolver um problema de encontrar automorfismos nÃo triviais de grafos. A validaÃÃo deste estado à feita atravÃs da aplicaÃÃo do algoritmo quÃntico de busca de Grover. Por fim à mostrado que a probabilidade de falsificaÃÃo da assinatura à negligenciÃvel dado o nÃmero de cÃpias do estado da assinatura.

The graph isomorphism problem has received a great deal of attention on both theoretical and practical fronts. However, a polynomial algorithm for the problem has yet to be found. Even so, the best of the existing algorithms perform well in practice; so well that it is challenging to find hard instances for them. The most efficient algorithms, for determining if a pair of graphs are isomorphic, are based on the individualization-refinement paradigm, pioneered by Brendan McKay in 1981 with his algorithm nauty. Nauty and various improved descendants of nauty, such as bliss and saucy, solve the graph isomorphism problem by determining a canonical representative for each of the graphs. The graphs are isomorphic if and only if their canonical representatives are identical. These algorithms also detect the symmetries in a graph which are used to speed up the search for the canonical representative--an approach that performs well in practice. Yet, several families of graphs have been shown to exist which are hard for nauty-like algorithms. This dissertation investigates why these graph families pose difficulty for individualization-refinement algorithms and proposes several techniques for circumventing these limitations. The first technique we propose addresses a fundamental problem pointed out by Miyazaki in 1993. He constructed a family of colored graphs which require exponential time for nauty (and nauty's improved descendants). We analyze Miyazaki's construction to determine the source of difficulty and identify a solution. We modify the base individualization-refinement algorithm by exploiting the symmetries discovered in a graph to guide the search for its canonical representative. This is accomplished with the help of a novel data structure called a guide tree. As a consequence, colored Miyazaki graphs are processed in polynomial time--thus obviating the only known exponential upper-bound on individualization-refinement algorithms (which has stood for the last 16 years). The preceding technique can only help if a graph has enough symmetry to exploit. It cannot be used for another family of hard graphs that have a high degree of regularity, but possess few actual symmetries. To handle these instances, we introduce an adaptive refinement method which utilizes the guide-tree data structure of the preceding technique to use a stronger vertex-invariant, but only when needed. We show that adaptive refinement is very effective, and it can result in dramatic speedups. We then present a third technique ideally suited for large graphs with a preponderance of sparse symmetries. A method was devised by Darga et al. for dealing with these large and highly symmetric graphs, which can reduce runtime by an order of magnitude. We explain the method and show how to incorporate it into our algorithm. Finally, we develop and implement a parallel algorithm for detecting the symmetries in, and finding a canonical representative of a graph. Our novel parallel algorithm divides the search for the symmetries and canonical representative among each processor, allowing for a high degree of scalability. The parallel algorithm is benchmarked on the hardest problem instances, and shown to be effective in subdividing the search space.

This thesis explores basic representation theory of finite groups, covering basic definitions such as irreducible representations. The main part of the work focuses on finding irreducible representations of automorphism groups of simple graphs, in particular for graphs consisting of two identical odd cycle components by using the knowledge of the automorphism group of cycle graphs. Character theory is used to find the irreducible representations.

A Steiner triple system of order v, denoted STS(v), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this thesis we give necessary and sufficient conditions for the existence of a tricyclic STS(v) when one of the cycles is of length one. In this case, the STS(v) will contain a subsystem which admits an automorphism consisting of a fixed point and a single cycle. The subsystem is said to be 1-rotational.