Ramanujan graphs from simplicial complexes with few blockers

Abstract

We present a construction method for Ramanujan graphs from simplicial complexes with one or two blockers, using the corresponding multivariable functions, where blockers mean minimal elements of the complement of a simplicial complex. First, we completely find all the binary linear codes obtained from simplicial complexes with one or two blockers, producing exactly 23 families of few-weight binary linear codes and 4 families of self-orthogonal codes. Furthermore, their weight distributions are explicitly computed by using the corresponding multivariable functions. Then we construct 14 families of nonbipartite Ramanujan graphs from the few-weight binary linear codes found; their eigenvalues are computed by using the weight distributions of the corresponding codes. We emphasize that it is the first time that the simplicial complexes with a few blockers are used for construction of few-weight linear codes and the corresponding Ramanujan graphs. Moreover, we verify that our families of Ramanujan graphs are different from the previous families in terms of eigenvalues and regularity.