In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by Sipser-Spielman [SS96] (and Tanner [Tan81]), and their analysis. We also show how to obtain expanders with many unique neighbors using similar ideas. There were many exciting results on this topic recently, starting with Asherov-Dinur [AD23] and Hsieh-McKenzie-Mohanty-Paredes [HMMP23], who gave a similar construction of unique neighbor expander graphs, but using more sophisticated ingredients (such as almost-Ramanujan graphs) and a more involved analysis. Subsequent beautiful works of Cohen-Roth-TaShma [CRT23] and Golowich [Gol23] gave even stronger objects (lossless expanders), but also using sophisticated ingredients. The main contribution of this work is that we get much more elementary constructions of unique neighbor expanders and with a simpler analysis.

翻译：在本笔记中，我们从具有温和扩展性质的谱扩展图或组合扩展图出发，给出了唯一邻居扩展图（unique neighbor expander graphs）的非常简单的构造方法。这些构造及其分析是Sipser-Spielman [SS96]（以及Tanner [Tan81]）从扩展图构造LDPC纠错码的方案及其分析的简单变体。我们还展示了如何利用类似思想获得具有许多唯一邻居的扩展图。近期该领域涌现了许多令人振奋的成果，始于Asherov-Dinur [AD23]和Hsieh-McKenzie-Mohanty-Paredes [HMMP23]，他们给出了唯一邻居扩展图的类似构造，但使用了更复杂的工具（如近乎拉马努金图）和更复杂的分析。后续Cohen-Roth-TaShma [CRT23]和Golowich [Gol23]的优美工作给出了更强对象（无损扩展图），但同样使用了复杂工具。本文的主要贡献在于，我们以更初等的方式实现了唯一邻居扩展图的构造，并给出了更简洁的分析。