Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

Abstract

Let CC d,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let AC d,k be the largest order in an Abelian Cayley graph for given d and k. We show that CC d,2 ≥ 13 36 (d + 2)(d − 4) for any d = 6p − 2 where p is a prime such that p = 13, p ≡ 1 (mod 13), and− 3 where q is a prime power.

Large circulant graphs of fixed diameter and arbitrary degree

“…In the next section, we extend a result of Vetrík [7] to deduce new lower bounds for L − C (2) and R − C (2). In Section 3, we describe a direct product construction and use it to build large cyclic Cayley graphs of small diameter and arbitrarily large degree.…”

“…In this instance, the trivial lower bound on L − C (2) is 1/4 and the trivial upper bound on L + C (2) is 1/2. Vetrík [7] (building on Macbeth,Šiagiová &Širáň [4]) presents a construction that proves that L In this section, we begin by extending this result to yield bounds for L − C (2) and R − C (2). This argument can also be found in Lewis [3].…”

“…Then by the result of Vetrík [7] we can construct a circulant graph of degree d , diameter 2 and order n = Since…”

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.…”

Cayley Graphs of Diameter Two from Difference Sets

“…proved in that there is always a prime p such that for sufficiently large x , we may extend to all sufficiently large integers d , by simply adding more elements into the corresponding generating set; see . Based on a similar approach, in Vetrík showed that for any , where p is a prime such that and .…”