3 research outputs found
Large butterfly Cayley graphs and digraphs
We present families of large undirected and directed Cayley graphs whose
construction is related to butterfly networks. One approach yields, for every
large and for values of taken from a large interval, the largest known
Cayley graphs and digraphs of diameter and degree . Another method
yields, for sufficiently large and infinitely many values of , Cayley
graphs and digraphs of diameter and degree whose order is exponentially
larger in than any previously constructed. In the directed case, these are
within a linear factor in of the Moore bound.Comment: 7 page
Large Cayley graphs of small diameter
The degree-diameter problem seeks to find the largest possible number of vertices in a graph having given diameter and given maximum degree. Very often the problem is studied for restricted families of graph such as vertex-transitive or Cayley graphs, with the goal being to find a family of graphs with good asymptotic properties. In this paper we restrict attention to Cayley graphs, and study the asymptotics by fixing a small diameter and constructing families of graphs of large order for all values of the maximum degree. Much of the literature in this direction is focused on the diameter two case. In this paper we consider larger diameters, and use a variety of techniques to derive new best asymptotic constructions for diameters 3, 4 and 5 as well as an improvement to the general bound for all odd diameters. Our diameter 3 construction is, as far as we know, the first to employ matrix groups over finite fields in the degree-diameter problem