Antipodal Distance Transitive Covers of Complete Graphs

AbstractA distance-transitive antipodal cover of a complete graphKnpossesses an automorphism group that acts 2-transitively on the fibres. The classification of finite simple groups implies a classification of finite 2-transitive permutation groups, and this allows us to determine all possibilities for such a graph. Several new infinite families of distance-transitive graphs are constructed

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties

Analysis And Construction Of Extremal Circulant And Other Abelian Cayley Graphs

This thesis concerns the analysis and construction of extremal circulant and other Abelian Cayley graphs. For the purpose of this thesis, extremal graphs are understood as graphs with largest possible order for given degree and diameter, and the search for them is called the degree-diameter problem. The emphasis is on circulant graphs and on families of graphs defined for infinite diameter classes for given fixed degrees. Most studies in the degree-diameter problem have employed candidate algebraic structures to generate graphs that successively improve on previous best results. In contrast, this study has made extensive use of computer searches to find extremal graphs and graph families directly, and has then sought the algebra that describes them. In this way, the maximum degree for which largest-known circulant graph families have been discovered, with order greater than the legacy lower bound, has been increased from 7 to 20 and beyond. Topics covered include graphs in the following categories, undirected unless stated otherwise: circulant, other Abelian Cayley, bipartite circulant, arc-transitive circulant, directed circulant and mixed circulant; and their main properties such as distance partition, odd girth and automorphism group size. A major aspect of this thesis is the analysis of a matrix associated with each graph family, the lattice generator matrix, with newly discovered properties such as quasimaximality, radius maximality and eccentricity. Important new relationships between graph families of common dimension have also been discovered: translation, conjugation and transposition. An Extremal Order Conjecture is established for extremal undirected circulant and other Abelian Cayley graphs of any degree and diameter. An equivalent conjecture for directed circulant graphs and certain classes of mixed circulants is also established. Most of the extremal and largest-known graphs and graph families presented here have been discovered by the author and are documented comprehensively in the appendices