Large Cayley graphs and vertex‐transitive non‐Cayley graphs of given degree and diameter

Abstract

For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3) k . By comparison with other available results in this area we show that our family gives the

Algebraic and computer-based methods in the undirected degree/diameter problem - A brief survey

“…Additional constructions of large Cayley graphs are given in [21,87,88,65,66,89,96,97,98]. Some of them also use combinations of direct and semidirect products of groups (e.g.…”

“…Some of them also use combinations of direct and semidirect products of groups (e.g. [87,65,96,97]). More information about Cayley graph constructions can be found in [71].…”

“…Recent progress in finding large graphs with given degree and diameter via voltage assignment includes [88,65,66,16], which illustrate these alternatives. For instance, [88] uses abelian groups on the dipole, while [65] uses non-abelian groups that can be factorized as a combination of direct and semidirect products. In [16], Canale and Rodríguez improve some of the values in the table of general graphs using a computer-guided search similar to that of [59].…”

On cubic non‐Cayley vertex‐transitive graphs

“…In 1983, the second author 22 asked for which positive integers n there exists a non‐Cayley vertex‐transitive graph on n vertices. Several articles directly or indirectly related to this subject (see 1, 2, 6, 13, 17, 19, 23–27, 29, 30, 32, 33, 36, 38, 39, 44for some of the relevant references), have appeared in the literature, answering this question for particular positive integers. For example, in 23 it is proved that every vertex‐transitive graph of order p k , where p is an odd prime and k ≤3, is a Cayley graph.…”

Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

“…for sufficiently large d and k ≤ d. Dougherty and Faber [1] presented a number of results on Abelian Cayley graphs for small d and large k. Constructions of Cayley graphs of non-Abelian groups can be found for example in [4] and [5].…”