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

\[
\lambda_g = \sum_{s \in S} \zeta_n^{\psi(gs)} = \sum_{i=1}^k \left[ \sum_{s, Rs = I_i} \zeta_n^{\psi(gs)} \right].
\]

We remark that by Corollary 2.7, if $s \in R$ such that $Rs = I_i = Rx_i$ then $s$ has a unique representation of the form $s = \hat{u}x_i$ where $u \in (R/\text{Ann}_R(x_i))^\times$ and $\hat{u}$ is a fixed lift of $u$ to $R^\times$. With this presentation, we can write

\[
\sum_{s, Rs = I_i} \zeta_n^{\psi(gs)} = \sum_{u \in (R/\text{Ann}_R(x_i))^\times} \zeta_n^{\psi(gux_i)} = \sum_{u \in (R/\text{Ann}_R(x_i))^\times} \zeta_n^{\psi_{x_i}(gu)} = c(g, R/\text{Ann}_R(x_i)).
\]

Here we recall that $\psi_{x_i}$ is the induced linear functional on $R/\text{Ann}_R(x_i)$. We conclude that $\lambda_g = \sum_{i=1}^k c(g, R/\text{Ann}_R(x_i))$. \( \square \)

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

**Corollary 4.17.** Suppose that $g' = ug$ for some $u \in R^\times$. Then $\lambda_g = \lambda_{g'}$.

**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 Ján Mináč for his constant encouragement and support.

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