# Graphs with circulant distance matrices

The cycle has this property. For instance, the distance matrix for a 6-cycle is:

$A=\begin{bmatrix} 0 & 1 & 2 & 3 & 2 & 1 \\\\ 1 & 0 & 1 & 2 & 3 & 2 \\\\ 2 & 1 & 0 & 1 & 2 & 3 \\\\ 3 & 2 & 1 & 0 & 1 & 2 \\\\ 2 & 3 & 2 & 1 & 0 & 1 \\\\ 1 & 2 & 3 & 2 & 1 & 0 \\\\ \end{bmatrix}$

The question is: is the cycle the only graph with this property?

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Other cyclic like graphs work, e.g. the complete graph, as well as complements of such graphs. Gerhard "Ask Me About System Design" Paseman, 2013.01.25 –  Gerhard Paseman Jan 25 '13 at 15:19