Cubic polyhedral Ramanujan graphs with face size no larger than six

Abstract

A k-regular graph is Ramanujan if its second largest eigenvalue (by magnitude) has magnitude less than or equal to \({2\sqrt{k-1}}\). Exhaustive search up to the bound of 138 vertices, derived from Spielman and Teng’s work on graph partitioning, finds all cubic polyhedral Ramanujan graphs with positive curvature, i.e., with face sizes no larger than 6. Of all such polyhedra, those with face sizes 5 or 6, i.e., fullerenes, give the largest known examples of cubic Ramanujan polyhedra (with 84 vertices). We also consider the notions of negative and positive Ramanujan graphs, as those without eigenvalues in the respective open intervals \({(-k,-2\sqrt{k-1})}\) and \({(2\sqrt{k-1},k)}\). Our results give the full list of positive cubic polyhedral Ramanujan graphs with positive curvature but for negative Ramanujan graphs we have only a finiteness theorem and a conjectured complete list.