Large Networks of Diameter Two Based on Cayley Graphs

In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we construct Cayley graphs of diameter two and of order greater than $\frac23$ of Moore bound and we show that Cayley graphs of degrees $d\in\{16,17,18,23,24,31,\dots,35\}$ constructed in this paper are the largest currently known vertex-transitive graphs of diameter two.Comment: 9 pages, Published in Cybernetics and Mathematics Applications in Intelligent System

Large graphs of diameter two and given degree

Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥ $d^2$ − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d + 1)$^2$/2 for degrees d = 2q − 1 where q is an odd prime power, and AC(d, 2) ≥ (3/8)($d^2$ − 4) where d = 4q − 2 for an odd prime power q. Using a number theory result on distribution of primes we prove, for all sufficiently large d, lower bounds on r(d, 2), C(d, 2), and AC(d, 2) of the form c · $d^2$ − O($d^1.525$) for c = 1, 1/2, and 3/8, respectively. We also prove results of a similar flavour for vertex transitive graphs and Cayley graphs of cyclic groups.Peer Reviewe

Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey

This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter

A note on large Cayley graphs of diameter two and given degree

For a variety of infinite sets of positive integers d related to odd prime powers we describe a simple construction of Cayley graphs of diameter two and given degree d which have order close to d2/2

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

Peer Reviewe