Graphs of given degree and diameter obtained as abelian lifts of dipoles

Abstract

a b s t r a c tWe derive an upper bound on the number of vertices in regular graphs of given degree and diameter arising as regular coverings of dipoles over abelian groups.

PolarStar: Expanding the Scalability Horizon of Diameter-3 Networks

“…The formal problem of designing a network 𝐺 with the largest order (number of nodes) for a given diameter 𝐷 and router degree 𝑑 is captured by the degree-diameter problem from [32]. The order of 𝐺 is bounded above by the Moore bound [19].…”

“…The mathematical problem of establishing the largest diameter-3 graphs is an open problem. There is a big gap between the bestknown diameter-3 graphs and the Moore bound for diameter 3 [32]. In this paper, we extend the orders of the largest known diameter-3 graphs and design a network based on these.…”

“…Best known Order in [32] Moore-bound Efficiency • A novel graph construction is presented here for the first time, giving the largest known graphs of diameter-3 for degrees 18−20 as per the Combinatorics Wiki leaderboard [32] (Table 1), superseding former records for the first time since 2010 on this open problem in an active field of mathematics. (The Wiki shows graphs only up to degree 20, so this construction likely gives "best" graphs at higher degrees as well).…”

“…Few graphs even come close to the Moore bound. The latest leaderboard of degree-diameter problem from [32] shows that the best known diameter-3 graphs, with the most relevant degrees, reach only 25−30% of the Moore bound. The PolarStar construction proposed here is larger than the best graphs for degrees 18 − 20, the highest degrees in the leaderboard, as shown in Table 1.…”

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

“…Just as in the case of Cayley graphs, recent results show that in order to obtain good graphs by voltage assignment, the underlying group should be as far as possible from abelian [60,32,33]. An alternative for the use of non-abelian groups is to use base graphs that are not bouquets (e.g.…”