On the Periodicity of k-distance Graphs

The $k$-distance graph $G^k$ of a graph $G$ has the same vertex set as $G$ and two vertices are adjacent if and only if their distance is $k$ in $G$. These graphs have been extensively studied for their connection properties. In this paper, we study various properties of these graphs, including clique number and periodicity. For all $k$, we show that there exists a $k$-distance graph that is weakly periodic for any size period. Our main result is a proof that if $k>2$ then there exists a $k$-distance graph of any size (strong) period. Finally, we provide evidence that there exists an upper bound on the periodicity of a connected $2$-distance graph as a function of $\mathrm{\mid }V\mathrm{\mid }$.