Expander Graphs -- Both Local and Global

Let $G=(V,E)$ be a finite graph. For $v\in V$ 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 $v\in V$. 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 $\{G_v|v\in V\}$ 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.