121 research outputs found

    Some Implications on Amorphic Association Schemes

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    AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition

    Some Implications on Amorphic Association Schemes

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    AMS classifications: 05E30, 05B20;

    Between primitive and 2-transitive : synchronization and its friends

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    The second author was supported by the Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through the project CEMAT-CIÊNCIAS UID/Multi/ 04621/2013An automaton (consisting of a finite set of states with given transitions) is said to be synchronizing if there is a word in the transitions which sends all states of the automaton to a single state. Research on this topic has been driven by the Černý conjecture, one of the oldest and most famous problems in automata theory, according to which a synchronizing n-state automaton has a reset word of length at most (n − 1)2 . The transitions of an automaton generate a transformation monoid on the set of states, and so an automaton can be regarded as a transformation monoid with a prescribed set of generators. In this setting, an automaton is synchronizing if the transitions generate a constant map. A permutation group G on a set Ω is said to synchronize a map f if the monoid (G, f) generated by G and f is synchronizing in the above sense; we say G is synchronizing if it synchronizes every non-permutation. The classes of synchronizing groups and friends form an hierarchy of natural and elegant classes of groups lying strictly between the classes of primitive and 2-homogeneous groups. These classes have been floating around for some years and it is now time to provide a unified reference on them. The study of all these classes has been prompted by the Černý conjecture, but it is of independent interest since it involves a rich mix of group theory, combinatorics, graph endomorphisms, semigroup theory, finite geometry, and representation theory, and has interesting computational aspects as well. So as to make the paper self-contained, we have provided background material on these topics. Our purpose here is to present recent work on synchronizing groups and related topics. In addition to the results that show the connections between the various areas of mathematics mentioned above, we include a new result on the Černý conjecture (a strengthening of a theorem of Rystsov), some challenges to finite geometers (which classical polar spaces can be partitioned into ovoids?), some thoughts about infinite analogues, and a long list of open problems to stimulate further work.PostprintPeer reviewe

    Searching for incomplete self orthogonal latin squares : a targeted and parallel approach

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    The primary purpose of this dissertation is in the search for new methods in which to search for Incomplete Self Orthogonal Latin Squares. As such a full understanding of the structures involved must be examined, starting from basic Latin Squares. The structures will be explained and built upon in order to cover Mutually Orthogonal Latin Squares, Frame Latin Squares and Self Orthogonal Latin Squares. In addition the related structure Orthogonal Arrays, will be explained as they relate to Incomplete Self Orthogonal Latin Squares. This paper also dedicates time to explaining basic search methods and optimizations that can be done. The two search methods of focus are the backtracking algorithm and heuristic searches. In our 6nal method the two will work together to achieve an improved result. The methods currently being used to search in parallel are also provided, along with the necessary backup to there structure. The main research of this paper is focused on the search for Incomplete Self Orthogonal Squares. This is done by breaking down the problem into four separate areas of the square. By separating the blocks it enables us to work on a smaller problem while eliminating many incorrect solutions. The solution methodology is broken up into three steps and systematically solving the individual areas of the square. By taking advantage of the properties of squares to constrain our search as much as possible we succeeded in reducing the total search time significantly. Unfortunately, even with our improvement in the overall search time, no open incomplete self orthogonal latin square problems could be solved. Full results and comparisons to existing methods are provided

    Glosarium Matematika

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    273 p.; 24 cm

    Glosarium Matematika

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    On strongly regular graphs

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    Strongly regular graphs are regular graphs with the additional property that the number of common neighbours for two vertices depends only on whether the vertices are adjacent or non-adjacent. From an algebraic point of view, a graph is strongly regular if its adjacency matrix has exactly three eigenvalues. Strongly regular graphs have very interesting algebraic properties due to their strong regularity conditions. Many strongly regular graphs are known to have large and interesting automorphism groups [23]. In [23] it is also conjectured that almost all strongly regular graphs are asymmetric. Peter Cameron in [7] mentions that "Strongly regular graphs lie on the cusp between highly structured and unstructured." Although strongly regular graphs have been studied extensively since they were introduced, there is very little known about the automorphism group of an arbitrary strongly regular graph based on its parameters. In this thesis, we have developed theory for studying the automorphisms of strongly regular graphs. Our study is both mathematical and computational. On the computational side, we introduce the notion of orbit matrices. Using these matrices, we were able to show that some strongly regular graphs do not admit an automorphism of a certain order. Given the size of the automorphism, we can generate all of the orbit matrices, using a computer program. Another computer program is implemented that generates all the strongly regular graphs from that orbit matrix. From a mathematical point of view, we have found an upper bound on the number of fixed points of the automorphisms of a strongly regular graph. This upper bound is a new upper bound and is obtained by algebraic technique
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