Asymptotic comparison of two constructions for large digraphs of given degree and diameter

summary:We compare the asymptotic growth of the order of the digraphs arising from a construction of Comellas and Fiol when applied to Faber-Moore digraphs versus plainly the Faber-Moore digraphs for the corresponding degree and diameter

Large digraphs of given diameter and degree from coverings

We show that a construction of Comellas and Fiol for large vertex-transitive digraphs of given degree and diameter from small digraphs preserves the properties of being a Cayley digraph and being a regular covering.Peer Reviewe

Extremal Metric and Topological Properties of Vertex Transitive and Cayley Graphs

We shall consider problems in two broad areas of mathematics, namely the area of the degree diameter problem and the area of regular maps. In the degree diameter problem we investigate finding graphs as large as possible with a given degree and diameter. Further, we may consider additional properties of such extremal graphs, for example restrictions on the kinds of symmetries that the graph in question exhibits. We provide two pieces of research relating to the degree diameter problem. First, we provide a new derivation of the Hoffman-Singleton graph and show that this derivation may be used with minor modification to derive the Bosák graph. Ultimately we show that no further natural modification of the construction we use can derive any other Moore or mixed-Moore graphs. Second, we answer the previously open question of whether the Gómez graphs, which are known to be vertex-transitive, are in addition also Cayley. In doing this, we also generalise the construction of the Gómez graphs and show that the Gómez graphs are the largest graphs for given degree and diameter following the generalised construction. We also provide two pieces of research relating to regular maps. We aim to address the related questions of for which triples of parameters k, l and m there exist finite regular maps of face length k, vertex order l and Petrie walk length m. We then address the related question of determining for which n there exist regular maps which are self dual and self Petrie dual which have face length, vertex order and Petrie dual walk length n. We address both questions by constructions of regular maps in fractional linear groups, necessarily leading us to study some interesting related number theoretic questions

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

Peer Reviewe