Let G = ( V, E ) be a finite graph. For v ∈ V we denote by G v the subgraph of G that is induced by v ’s neighbor set. We say that G is ( a,b )-regular for a > b > 0 integers, if G is a -regular and G v is b-regular for every v ∈ V . Recent advances in PCP theory call for the construction of infinitely many ( a,b )-regular expander graphs G that are expanders also locally. Namely, all the graphs { G v ∣ v ∈ V } should be expanders as well. While random regular graphs are expanders with high probability, they almost surely fail to expand locally. Here we construct two families of ( a,b )-regular graphs that expand both locally and globally. We also analyze the possible local and global spectral gaps of ( a,b )-regular graphs. In addition, we examine our constructions vis-a-vis properties which are considered characteristic of high-dimensional expanders.

中文翻译：

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扩展器图 - 本地和全局

令 G = ( V, E ) 为有限图。对于 v ∈ V，我们用 G v 表示 G 的子图，它是由 v 的邻居集诱导的。我们说 G 对于 a > b > 0 整数是 ( a,b )-regular ，如果 G 是 a -regular 并且 G v 对于每个 v ∈ V 都是 b-regular 。PCP 理论的最新进展要求构建无限多个 (a,b)-正则展开图 G，这些图 G 也是局部展开图。即，所有的图 { G v ∣ v ∈ V } 也应该是扩展器。虽然随机正则图是很有可能的扩展器，但它们几乎肯定无法局部扩展。在这里，我们构建了两个 (a,b) 正则图族，它们在局部和全局上都可以扩展。我们还分析了 (a,b)-正则图可能的局部和全局光谱间隙。此外，