Tight Bound on the Diameter of the Knödel Graph

Abstract

The Knödel graph $W_{\varDelta,n}$ is a regular graph of even order and degree $\varDelta$ where 2 $\leq \varDelta \leq \lfloor{log_2 n}\rfloor$. Despite being a highly symmetric and widely studied graph, the diameter of $W_{\varDelta,n}$ is known only for $n=2^{\varDelta}$. In this paper we present a tight upper bound on the diameter of the Knödel graph for general case. We show that the presented bound differs from the diameter by at most 2 when $\varDelta < \alpha \lfloor{\log_2 n}\rfloor$ for some 0 < α < 1 where α → 1 when n → ∞. The proof is constructive and provides a near optimal diametral path for the Knödel graph $W_{\varDelta,n}$.