Constructing Ramanujan Graphs Using Shift Lifts

In a breakthrough work, Marcus et al. [15] recently showed that every d-regular bipartite Ramanujan graph has a 2-lift that is also d-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a d-regular bipartite Ramanujan graph on N vertices in time 2. Shift k-lifts studied in [1] lead to a natural approach for constructing Ramanujan graphs more efficiently. The number of possible shift k-lifts of a d-regular n-vertex graph is k. Suppose the following holds for k = 2: There exists a shift k-lift that maintains the Ramanujan property of d-regular bipartite graphs on n vertices for all n. (?) Then, by performing a similar brute-force algorithm, one would be able to construct an N -vertex bipartite Ramanujan graph in time 2 log 2 . Also, if (?) holds for all k ≥ 2, then one would obtain an algorithm that runs in polyd(N) time. In this work, we take a first step towards proving (?) by showing the existence of shift k-lifts that preserve the Ramanujan property in d-regular bipartite graphs for k = 3, 4.