Analytic methods for uniform hypergraphs∗†

This paper presents some analytic methods for studying uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues λ and λmin of 2-graphs. First, these two parameters are extended to weighted uniform hypergraphs; second, the eigenvalues-numbers λ and λmin are extended to eigenvalues-functions λ(p) and λ (p) min, which also encompass other graph parameters like the Lagrangian and the number of edges. In this way the functions λ(p) and λ (p) min seamlessly join spectral and traditional results in hypergraphs. In particuler, this new viewpoint helps to show that spectral extremal and edge extremal problems are asymtotically equivalent. Naturally, all results about λ(p) and λ (p) min also extend spectral hypergraph theory, but delve into deeper problems than before. In fact, the resulting theory is new even for 2graphs, where some well-settled topics become research challenges again. The paper covers a multitude of topics, with more than a hundred concrete statements to underpin an analytic theory for hypergraphs. Essential among these topics are a PerronFrobenius type theory and methods for extremal hypergraph problems. Many open problems are raised and directions for possible further research are outlined. ∗AMS MSC: 05C65; 05C35. †