Expander Graphs — Both Local and Global

Abstract

Let G = (V, E) be a finite graph. For v ∈ V we denote by Gv 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 Gv 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 {Gv ∣ 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.