On the Spectral Gap and the Diameter of Cayley Graphs

Abstract

We obtain a new bound connecting the first nontrivial eigenvalue of the Laplace operator on a graph and the diameter of the graph. This bound is effective for graphs with small diameter as well as for graphs with the number of maximal paths comparable to the expected value.

A spectral bound for vertex-transitive graphs and their spanning subgraphs

“…We obtain the following result for 'large' spanning subgraphs of vertex-transitive graphs, thus establishing an analogue of the discrete Cheeger-Buser inequality for these graphs. For a bound on the second largest eigenvalue in terms of the diameter and the degree, we refer to the works of Saloff-Coste [27, Corollary 3.2.7], Shkredov [28].…”

On multiplicative energy of subsets of varieties

“…Even in the abelian case, it is easy to see that not each Bohr set is regular (see, e.g., [35,Section 4.4]). Nevertheless, it can be showed (see, e.g., [32]) that one can find a regular Bohr set decreasing the parameter δ slightly.…”

“…Let us remark an universal lower bound for the size of any Bohr set (see [26,Lemma 17.3], [32,Proposition 28] for the case of multidimensional Bohr sets).…”

“…Denote by the operator of convex conjugation. As in [31], define Put . For example, we have, clearly, .…”

“…For example, we have, clearly, . It is easy to see that defines a norm of a function f (see [31]). This fact follows from the following inequality [31, Lemma 10]: In particular, .…”