A new approach for computing the distance and the diameter in circulant graphs

The diameter of a graph is the maximum distance among all pairs of vertices. Thus a graph $G$ has diameter $d$ if any two vertices are at distance at most $d$ and there are two vertices at distance $d$. We are interested in studying the diameter of circulant graphs $C_n(1,s)$, i.e., graphs with the set $\{0,1,\ldots, n-1\}$ of integers as vertex set and in which two distinct vertices $i,j \in \{0,1,\ldots, n-1\}$ are adjacent if and only if $|i-j|_n\in \{1,s\}$, where $2\leq s\leq \lfloor \frac{n-1}{2} \rfloor$ and $|x|_n=\min(|x|, n-|x|)$. Despite the regularity of circulant graphs, it is difficult to evaluate several parameters, in particular the distance and the diameter. To the best of our knowledge, there is no formulas providing exact values for the distance and the diameter of $C_n(1,s)$ for all $n$ and $s$. In this context, we present in this paper a new approach, based on a simple algorithm, that gives exact values for the distance and the diameter of circulant graphs.