Second Talk on Ramanujan Graphs

We are especially interested in graphs – or better, in sequences of graphs – whose first eigenvalues λ1 are relatively small. In particular, for any finite graph, define the spectral gap ω(G) = λ0(G) − λ1(G). Define also the isoperimetric constant h(G) to be the infimum #E(V1, V2) min{#V1, #V2} over all partitions of the vertex set into two subsets V1, V2; here E(V1, V2) is the set of edges connecting a vertex in V1 to a vertex in V2. The larger h is the “better connected” G is as a network. An expander graph is a sequence of finite graphs Gn with #Gn →∞ (equivalently, any fixed isomorphism type occurs only finitely many times in the sequence) and inf h(Gn) > 0. It turns out to be the case that this condition is equivalent to inf ω(Gn) > 0 (e.g. [2]).