Let R be a finite commutative ring with non-zero identity. Let $R^{\times }$ and $J\left(R\right)$ be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by $G\left(R\right)$, is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if $x+y\in R^{\times }$. If we relax this definition by dropping the term ‘distinct’, we obtain the closed unit graph, denoted by $\overline{G}\left(R\right)$. In this paper, we compute the adjacency spectrum of the graph $\overline{G}\left(R\right)$. We utilize this result to show that $G\left(R\right)\cong G\left(S\right)$ if and only if $\left(R/J\left(R\right)\right)\cong \left(S/J\left(S\right)\right)$ and $|J\left(R\right)|=|J\left(S\right)|$, where R and S are two arbitrary finite rings. Moreover, we determine when $G\left(R\right)$ is a Ramanujan graph. We also deliver a necessary and sufficient condition for $G\left(R\right)$ to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs.

令R为非零单位的有限交换环。让$R^{\times }$和$Ĵ\left(R\right)$分别是单元元素群和R的 Jacobson 根。环R的单位图，记为$G\left(R\right)$, 是一个图，其顶点集是R并且两个不同的顶点x和y是相邻的当且仅当$X+\mathrm{是的}\in R^{\times }$. 如果我们通过删除术语“distinct”来放宽这个定义，我们会得到封闭的单位图，表示为$\overline{G}\left(R\right)$. 在本文中，我们计算图的邻接谱$\overline{G}\left(R\right)$. 我们利用这个结果来证明$G\left(R\right)\cong G\left(\mathrm{小号}\right)$当且仅当$\left(R/Ĵ\left(R\right)\right)\cong \left(\mathrm{小号}/Ĵ\left(\mathrm{小号}\right)\right)$和$|Ĵ\left(R\right)|=|Ĵ\left(\mathrm{小号}\right)|$，其中R和S是两个任意有限环。此外，我们确定何时$G\left(R\right)$是拉马努金图。我们还提供了一个充分必要条件$G\left(R\right)$成为一个强正则图。最后，我们获得了单位凯莱图和酉凯莱图的泛化谱。