Cyclic Cayley Graphs

From CanisiusmathWiki

This page has my preliminary observations about the Cayley Graphs of cyclic groups.

Spectrum

In Julia Lazenby's thesis Circulant Graphs and Their Spectra she notes that the eigen values of $Cay(Z_n,\{S\})$ are
$\lambda_x = \sum_{s \in S}e^{2 \pi i x s/ n}$ where $x: 1, 2, ... n$
or more simply
$\lambda_x = \sum_{s \in S}\omega_{n}^{x s}$ where $x: 1, 2, ... n$
In order to get the spectrum of the quotient groups of $Z_n$ using the same roots of unity we have to modify the equation to include a change to a different set of roots of unity.

Let $Z_n$ be a group
Let $Z_m$ be a quotient group of $Z_n$
Then the eigen values of $Cay(Z_m,\{S\})$ are
$\lambda_x = \sum_{s \in S}\omega_{n}^{(n/m)(x s\mod m)}$ where $x: 1, 2, ... m$