$\text{Proof. Let } \mathcal{S} \text{ be the generating set associated with } D \text{ as described in Proposition 2.5. By} $  $\text{the circulant diagonalization theorem, the spectrum of } G\_\mathbb{R}(D) = \Gamma(\mathbb{R}, \mathbb{S}) \text{ is the multiset} $  $\{\lambda\_{\mathcal{S}}\}\_{\mathcal{S}\in\mathcal{R}}\text{ where} $ 

$$\lambda\_{\mathcal{S}} = \sum\_{\mathbf{s} \in \mathcal{S}} \mathfrak{J}\_n^{\psi(\mathcal{gs})} = \sum\_{i=1}^k \left[ \sum\_{\mathbf{s}, \mathbf{R} \mathbf{s} = \mathcal{Z}\_i} \mathfrak{J}\_n^{\psi(\mathcal{gs})} \right].$$

 $\text{We remark that by Corollary 2.7, if } \mathbf{s} \in \mathbb{R} \text{ such that } \mathbf{R}\mathbf{s} = \mathbb{Z}\_{\mathbf{i}} = \mathbf{R}\mathbf{x}\_{\mathbf{i}} \text{ then } \mathbf{s} \text{ has a unique} $  $\text{representation of the form } s = \mathfrak{l}x\_{\mathbb{I}} \text{ where } u \in (\mathbb{R}/\text{Ann}\_{\mathbb{R}}(x\_{\mathbb{I}}))^{\times} \text{ and } \mathfrak{l} \text{ is a fixed lift of } u $  $\text{to } \mathbb{R}^\times. \text{ With this presentation, we can write} $ 

$$\sum\_{s, \mathbf{R} = \mathbb{Z}\_i} \zeta\_n^{\psi(\mathbf{g}s)} = \sum\_{u \in (\mathbb{R}/\mathbf{Ann}\_\mathbb{R}(\mathbf{x}\_l))^\times} \zeta\_n^{\psi(\mathbf{gux}\_l)} = \sum\_{u \in (\mathbb{R}/\mathbf{Ann}\_\mathbb{R}(\mathbf{x}\_l))^\times} \zeta\_n^{\psi\_{\mathbf{x}\_l}(\mathbf{g}u)} = c(\mathbf{g}, \mathbb{R}/\mathbf{Ann}\_\mathbb{R}(\mathbf{x}\_l)).$$

 $\text{Here we recall that } \psi\_{\mathbf{x}\_i} \text{ is the induced linear functional on } \mathbb{R}/\text{Ann}\_{\mathbb{R}}(\mathbf{x}\_i). \text{ We conclude} $  $\text{that } \lambda\_{\mathcal{S}} = \sum\_{i=1}^{k} $ *c*(*g*, *R*/Ann*R*(*xi*)). □

The following corollary is simple yet important for our future work on perfect state transfers on gcd-graphs.

 $\textbf{Corollary 4.17.}\text{ Suppose that }\mathsf{g'} = \mathsf{ug}\text{ for some }\mathsf{u}\in\mathsf{R}^\times.\text{ Then }\lambda\_{\mathcal{S}} = \lambda\_{\mathcal{g'}}. $ 

## ACKNOWLEDGEMENTS

We thank the Department of Mathematics and Computer Science at Lake Forest College for their generous financial support through an Overleaf subscription. We also thank Jan Min ´ a´c for his constant encouragement and support. ˇ

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