The expansion rate of Margulis expanders and LPS expanders for vertex set Z × Z

Expanders are potential mathematic tools to design robust computer networks and to analyze the formation and expansion of today's Internet. In this paper, we derive lower bounds regarding the expansion rate of Margulis expanders, lower bounds and upper bounds regarding the expansion rate of LPS expanders, for vertex set Z × Z . The two types of expanders we study are variants of the expanders due to Margulis [4] and Lubotzky, Philips, Sarnak (LPS) [3] respectively. We consider the compact planar integer point-sets Z × Z and the corresponding counting measure. Following [1], the proof is based on symmetrization.