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A circulant matrix is a square matrix where each row has the same elements as the previous row, cyclically rotated right by one element. It is a specific kind of a square Toeplitz matrix.
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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 \\\\
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bounds on the entries of an inverse circulant matrix
Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the entri …