The order of the automorphism group of a binary qq-analog of the Fano plane is at most two

It is shown that the automorphism group of a binary $q$-analog of the Fano plane is either trivial or of order $2$.Comment: 10 page

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe

Generation of Graph Classes with Efficient Isomorph Rejection

In this thesis, efficient isomorph-free generation of graph classes with the method of generation by canonical construction path(GCCP) is discussed. The method GCCP has been invented by McKay in the 1980s. It is a general method to recursively generate combinatorial objects avoiding isomorphic copies. In the introduction chapter, the method of GCCP is discussed and is compared to other well-known methods of generation. The generation of the class of quartic graphs is used as an example to explain this method. Quartic graphs are simple regular graphs of degree four. The programs, we developed based on GCCP, generate quartic graphs with 18 vertices more than two times as efficiently as the well-known software GENREG does. This thesis also demonstrates how the class of principal graph pairs can be generated exhaustively in an efficient way using the method of GCCP. The definition and importance of principal graph pairs come from the theory of subfactors where each subfactor can be modelled as a principal graph pair. The theory of subfactors has applications in the theory of von Neumann algebras, operator algebras, quantum algebras and Knot theory as well as in design of quantum computers. While it was initially expected that the classification at index 3 + √5 would be very complicated, using GCCP to exhaustively generate principal graph pairs was critical in completing the classification of small index subfactors to index 5¼. The other set of classes of graphs considered in this thesis contains graphs without a given set of cycles. For a given set of graphs, H, the Turán Number of H, ex(n,H), is defined to be the maximum number of edges in a graph on n vertices without a subgraph isomorphic to any graph in H. Denote by EX(n,H), the set of all extremal graphs with respect to n and H, i.e., graphs with n vertices, ex(n,H) edges and no subgraph isomorphic to any graph in H. We consider this problem when H is a set of cycles. New results for ex(n, C) and EX(n, C) are introduced using a set of algorithms based on the method of GCCP. Let K be an arbitrary subset of {C3, C4, C5, . . . , C32}. For given n and a set of cycles, C, these algorithms can be used to calculate ex(n, C) and extremal graphs in Ex(n, C) by recursively extending smaller graphs without any cycle in C where C = K or C = {C3, C5, C7, . . .} ᴜ K and n≤64. These results are considerably in excess of the previous results of the many researchers who worked on similar problems. In the last chapter, a new class of canonical relabellings for graphs, hierarchical canonical labelling, is introduced in which if the vertices of a graph, G, is canonically labelled by {1, . . . , n}, then G\{n} is also canonically labelled. An efficient hierarchical canonical labelling is presented and the application of this labelling in generation of combinatorial objects is discussed

Computer classification of linear codes

We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements. We validate a correct implementation of the algorithms with known classification results from the literature, which we partially extend to larger ranges of parameters.Comment: 18 pages, 9 tables; this paper is a merge and extension of arXiv:1907.10363 and arXiv:1912.0935

Automatic techniques for detecting and exploiting symmetry in model checking

The application of model checking is limited due to the state-space explosion problem – as the number of components represented by a model increase, the worst case size of the associated state-space grows exponentially. Current techniques can handle limited kinds of symmetry, e.g. full symmetry between identical components in a concurrent system. They avoid the problem of automatic symmetry detection by requiring the user to specify the presence of symmetry in a model (explicitly, or by annotating the associated specification using additional language keywords), or by restricting the input language of a model checker so that only symmetric systems can be specified. Additionally, computing unique representatives for each symmetric equivalence class is easy for these limited kinds of symmetry. We present a theoretical framework for symmetry reduction which can be applied to explicit state model checking. The framework includes techniques for automatic symmetry detection using computational group theory, which can be applied with no additional user input. These techniques detect structural symmetries induced by the topology of a concurrent system, so our framework includes exact and approximate techniques to efficiently exploit arbitrary symmetry groups which may arise in this way. These techniques are also based on computational group theoretic methods. We prove that our framework is logically sound, and demonstrate its general applicability to explicit state model checking. By providing a new symmetry reduction package for the SPIN model checker, we show that our framework can be feasibly implemented as part of a system which is widely used in both industry and academia. Through a study of SPIN users, we assess the usability of our automatic symmetry detection techniques in practice