The proper power graph $P^{\ast }\left(G\right)$ of a group G is the simple graph with non-trivial elements of G as vertices and two distinct elements are adjacent if and only if one is a power of the other. The aim of this paper is to find the structure and to compute the spectrum of the proper power graph of direct product of two finite groups $G_1$ and $G_2,$ where at least one of $G_1$ and $G_2$ is an EPO group. Also we provide the full spectrum and hence the energy of $P^{\ast }(G_1\times G_2)$ whenever both $G_1$ and $G_2$ are EPO groups.

中文翻译：

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某些有限群的直积的固有幂图的谱

正确的功率图$P^{＊}\left(G\right)$群G的一个简单图是以G的非平凡元素为顶点，并且两个不同的元素相邻当且仅当一个元素是另一个元素的幂时。本文的目的是寻找结构并计算两个有限群直积的本征幂图的谱$G_{\mathrm{1个}}$和$G_{\mathrm{2个}},$其中至少一个$G_{\mathrm{1个}}$和$G_{\mathrm{2个}}$是一个 EPO 组。我们还提供全光谱，因此提供能量$P^{＊}(G_{\mathrm{1个}}\times G_{\mathrm{2个}})$每当两者$G_{\mathrm{1个}}$和$G_{\mathrm{2个}}$是 EPO 组。