Ramanujan Graphs (XIII)

by Sanming Zhou

Institution: The University of Melbourne
Date: Wed 19th June 2013
Time: 11:00 AM
Location: Room 107, Richard Berry Building

Abstract: We will continue to discuss Chapter 4 of the book `Elementary Number Theory, Group Theory, and Ramanujan Graphs' (G. Davidoff, P. Sarnak and A. Valette, Cambridge University Press, 2003). We will introduce a family of connected Cayley graphs $Y^{p, q}$, give a lower bound on their girths, and prove that $Y^{p, q}$ is isomorphic to $X^{p, q}$. In this way we prove that the graphs $X^{p, q}$ are connected and have large girth. We may also talk about a few preliminary results required to estimate the spectral gap of these graphs.