Analytic methods for uniform hypergraphs

TL;DR

This paper develops new analytic methods for uniform hypergraphs, extending spectral theory from 2-graphs to hypergraphs, introducing eigenvalues-functions, and addressing extremal problems with numerous new results and open questions.

Contribution

It introduces a comprehensive analytic framework for hypergraphs, generalizing spectral concepts and eigenvalues-functions, and applies these to extremal hypergraph problems.

Findings

Extension of spectral theory to weighted r-graphs

Introduction of eigenvalues-functions encompassing graph parameters

Development of Perron-Frobenius type theory for hypergraphs

Abstract

This paper develops analityc methods for investigating uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple setup is extended to weighted r-graphs, and on the other, the eigenvalues-numbers are generalized to eigenvalues-functions, which encompass also other graph parameters like Lagrangians and number of edges. The resulting theory is new even for 2-graphs, where well-settled topics become 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 Perron-Frobenius type theory and methods for extremal hypergraph problems. Many open problems are raised and directions for possible further research are outlined.