Large "De Bruijn" Cayley graphs and digraphs (original) (raw)

Large Cayley graphs of small diameter

arXiv (Cornell University), 2017

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.

Abelian Cayley digraphs with asymptotically large order for any given degree

Abelian Cayley digraphs can be constructed by using a generalization to ZnZ^nZn of the concept of congruence in ZZZ. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have asymptotically large number of vertices as the diameter increases. Up to now, the best known asymptotically dense results were all non-constructive.

A Note on Constructing Large Cayley Graphs of Given Degree and Diameter by Voltage Assignments

The Electronic Journal of Combinatorics, 1998

Voltage graphs are a powerful tool for constructing large graphs (called lifts) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem, which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree ≤ 15 and diameter ≤ 10 have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups.

Approaching the Moore bound for diameter two by Cayley graphs

Journal of Combinatorial Theory, Series B, 2012

The order of a graph of maximum degree d and diameter 2 cannot exceed d 2 + 1, the Moore bound for diameter two. A combination of known results guarantees the existence of regular graphs of degree d, diameter 2, and order at least d 2 − 2d 1.525 for all sufficiently large d, asymptotically approaching the Moore bound. The corresponding graphs, however, tend to have a fairly small or trivial automorphism group and the nature of their construction does not appear to allow for modifications that would result in a higher level of symmetry. The best currently available construction of vertex-transitive graphs of diameter 2 and preassigned degree gives order 8 9 (d + 1 2 ) 2 for all degrees of the form d = (3q − 1)/2 for prime powers q ≡ 1 mod 4.

An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

Discrete Mathematics, 2020

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.

Asymptotic aspects of Cayley graphs

2011

Arising from complete Cayley graphs Γ n of odd cyclic groups Z n , an asymptotic approach is presented on connected labeled graphs whose vertices are labeled via equallymulticolored copies of K 4 in Γ n with adjacency of any two such vertices whenever they are represented by copies of K 4 in Γ n sharing two equally-multicolored triangles. In fact, these connected labeled graphs are shown to form a family of graphs of largest degree 6 and diameter asymptotically of order |V | 1/3 , properties shared by the initial member of a collection of families of Cayley graphs of degree 2m ≥ 6 with diameter asymptotically of order |V | 1/m , where 3 ≤ m ∈ Z.

Large bipartite Cayley graphs of given degree and diameter

Discrete Mathematics, 2011

Let BC d,k be the largest possible number of vertices in a bipartite Cayley graph of degree d and diameter k. We show that BC d,k ≥ 2(k − 1)((d − 4)/3) k−1 for any d ≥ 6 and any even k ≥ 4, and BC d,k ≥ (k − 1)((d − 2)/3) k−1 for d ≥ 6 and k ≥ 7 such that k ≡ 3 (mod 4).