Eigenvalues of non-regular linear quasirandom hypergraphs

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to coregular kk-uniform hypergraphs with loops. However, for k≥3k≥3 no kk-uniform hypergraph is coregular.In this paper we remove the coregular requirement. Consequently, the characterization can be applied to kk-uniform hypergraphs; for example it is used in Lenz and Mubayi (2015) [5] to show that a construction of a kk-uniform hypergraph sequence has some quasirandom properties. The specific statement that we prove here is that if a kk-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then its second largest eigenvalue is much smaller than its largest one.