Expander Graphs – Both Local and Global

Abstract

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.