Diameter of General Kn\"odel Graphs

The Kn"odel graph $W_{\mathrm{\Delta },n}$ is a $\mathrm{\Delta }$-regular bipartition graph on $n\ge 2^{\mathrm{\Delta }}$ vertices and $n$ is an even integer. The vertices of $W_{\mathrm{\Delta },n}$ are the pairs $(i,j)$ with $i=1,2$ and $0\le j\le n\mathrm{/}2-1$. For every $j$, $0\le j\le n\mathrm{/}2-1$, there is an edge between vertex $(1,j)$ and every vertex $(2,(j+2^k-1)\mathrm{m}\mathrm{o}\mathrm{d}(n\mathrm{/}2))$, for $k=0,1,\cdots ,\mathrm{\Delta }-1$. In this paper we obtain some formulas for evaluating the distance of vertices of the Kn"odel graph and by them, we provide the formula $diam(W_{\mathrm{\Delta },n})=1+\lceil \frac{n-2}{2^{\mathrm{\Delta }}-2}\rceil$ for the diameter of $W_{\mathrm{\Delta },n}$, where $n\ge (2\mathrm{\Delta }-5)(2^{\mathrm{\Delta }}-2)+4$.