Improved lower bounds on the degree–diameter problem

Abstract

Let C(d, k) and AC(d, k) be the largest order of a Cayley graph and a Cayley graph based on an abelian group, respectively, of degree d and diameter k. It is well known that $$C(d,k)\le 1+d+d(d-1)+\cdots +d(d-1)^{k-1}$$ with equality satisfied if and only if the graph is a Moore graph. However, there is a much better upper bound for abelian Cayley graph. We have $$AC(d,2)\le \frac{d^{2}}{2}+d+1$$ and $$AC(d,k)\le \frac{d^{k}}{k!}+O(d^{k-1})$$ . On the other hand, the best currently lower bounds are $$C(d,2)\ge 0.684d^{2}$$ , $$AC(d,2)\ge \frac{25}{64}d^{2}-2.1d^{1.525}$$ and $$AC(d,k)\ge (\frac{d}{k})^{k}+O(d^{k-1})$$ for sufficiently large d. In this paper, we improve previous results on the degree–diameter problem. We show that $$C(d,2)\ge \frac{200}{289}d^{2}-5.4d^{1.525}$$ , $$AC(d,2)\ge \frac{27}{64}d^{2}-3.9d^{1.525}$$ and $$AC(d,k)\ge (\frac{3}{3k-1})^{k}d^{k}+O(d^{k-0.475})$$ for sufficiently large d.