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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
Regular Embeddings of Canonical Double Coverings of Graphs
AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productGâK2, can be described in terms of regular embeddings ofG. This allows us to âliftâ the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the âderivedâ maps by employing those of the âbaseâ maps. We apply these results to determining all orientable regular embeddings of the tensor productsKnâK2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKnâK2exist only ifnis a prime powerpl, and there are 2Ï(nâ1) orÏ(nâ1) isomorphism classes of such maps (whereÏis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes
New methods for finding minimum genus embeddings of graphs on orientable and non-orientable surfaces
The question of how to find the smallest genus of all embeddings of a given finite connected
graph on an orientable (or non-orientable) surface has a long and interesting history.
In this paper we introduce four new approaches to help answer this question, in both the
orientable and non-orientable cases. One approach involves taking orbits of subgroups of
the automorphism group on cycles of particular lengths in the graph as candidates for subsets
of the faces of an embedding. Another uses properties of an auxiliary graph defined
in terms of compatibility of these cycles. We also present two methods that make use
of integer linear programming, to help determine bounds for the minimum genus, and to
find minimum genus embeddings. This work was motivated by the problem of finding the
minimum genus of the Hoffman-Singleton graph, and succeeded not only in solving that
problem but also in answering several other open questions
Reflexibility of regular Cayley maps for abelian groups
In this paper, properties of reflexible Cayley maps for abelian groups are investigated, and as a result, it is shown that a regular Cayley map of valency greater than 2 for a cyclic group is reflexible if and only if it is anti-balanced