We show that on every Ramanujan graph @math , the simple random walk exhibits cutoff: when @math has @math vertices and degree @math , the total-variation distance of the walk from the uniform distribution at time @math is asymptotically @math where @math is a standard normal variable and @math is an explicit constant. Furthermore, for all @math , @math -regular Ramanujan graphs minimize the asymptotic @math -mixing time for SRW among all @math -regular graphs. Our proof also shows that, for every vertex @math in @math as above, its distance from @math of the vertices is asymptotically @math .
A new impetus for understanding distances in Ramanujan graphs is due to their role as building blocks in quantum computing; see the influential letter by P. Sarnak @cite_25 . Some of Sarnak's ideas were developed further by his student N.T. Sardari in an insightful paper @cite_19 posted to the arXiv a few months after the initial posting of the present paper. For a certain infinite family of @math -regular @math -vertex Ramanujan graphs, Sardari @cite_19 shows that the diameter is at least @math and also gives an alternative proof of the first part of Corollary .
@cite_19 @cite_25
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For an infinite family of @math -regular LPS Ramanujan graphs, we show that the diameter of these graphs is greater than or equal to @math , where @math is an odd prime number and @math is the number of vertices. On the other hand, for any @math -regular Ramanujan graph we show that the distance of only tiny fraction of all pairs of vertices is greater than @math . We also have some numerical experiments for LPS Ramanujan graphs and random Cayley graphs which suggest that the diameters are asymptotically @math and @math , respectively. These are consistent with Sarnak's expectation on the covering exponent of universal quantum gates and our conjecture for the optimal strong approximation for quadratic forms in 4 variables. In this paper, we explore spectral properties of a class of regular Cayley graphs known as Ramanujan graphs and prove that the ratio of their algebraic connectivity to that of regular lattices grows exponentially as O(ngamma) with gamma = 1.84plusmn0.05 for networks with average degree of O(log(n)). Explicit construction algorithms exist for Ramanujan graphs that create regular graphs with especial degree and scale that depend on a pair of prime numbers. We introduce a randomized algorithm for construction of a class of fast regular graphs called quasi Ramanujan graphs. These graphs are obtained from finite number of degree balancing operations on Watts-Strogatz small-word networks that are irregular graphs. We show that quasi Ramanujan graphs share similar combinatorial optimality spectral properties as Ramanujan graphs and are not restricted to especial choices of degree and scale. A byproduct of this fact is that the algebraic connectivity ratio of quasi Ramanujan graphs grows exponentially in n as well. Numerical experiments are performed to verify our analytical predictions. Consensus algorithms converge extremely fast on networks with exponentially growing algebraic connectivity ratios. We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also establish the existence of infinite families of irregular Ramanujan' graphs, whose eigenvalues are bounded by the spectral radius of their universal cover. Such families were conjectured to exist by Linial and others. In particular, we prove the existence of infinite families of (c,d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by sqrt c-1 +sqrt d-1 , for all c, d 3. Our proof exploits a new technique for demonstrating the existence of useful combinatorial objects that we call the "method of interlacing polynomials'". We prove that any graph excluding Kr as a minor has can be partitioned into clusters of diameter at most Δ while removing at most O(r Δ) fraction of the edges. This improves over the results of Fakcharoenphol and Talwar, who building on the work of Klein, Plotkin and Rao gave a partitioning that required to remove O(r2 Δ) fraction of the edges. Our result is obtained by a new approach that relates the topological properties (excluding a minor) of a graph to its geometric properties (the induced shortest path metric). Specifically, we show that techniques used by Andreae in his investigation of the cops and robbers game on graphs excluding a fixed minor, can be used to construct padded decompositions of the metrics induced by such graphs. In particular, we get probabilistic partitions with padding parameter O(r) and strong-diameter partitions with padding parameter O(r2) for Kr-free graphs, O(k) for treewidth-k graphs, and O(log g) for graphs with genus g.