Spectral hypergraph theory via tensors

Abstract

The thesis is devoted to a few problems on spectral hypergraphs theory via the adjacency tensor, Laplacian tensor and signless Laplacian tensor of a hypergraph. These problems are analogy and generalization of problems usually concerned in spectral graph theory. Three topics are included: 1. Characterization of uniform hypergraphs with largest spectral radii under certain conditions. 2. The property of symmetric spectrum for uniform hypergraphs with applications. 3. Properties on the spectra of non-uniform and general hypergrpahs. For the first topic, two types of connected hypergraphs with fixed vertex number and cyclomatic number called unicyclic and bicyclic hypergraphs are studied. By combining recent developed spectral techniques, the first five hypergraphs with largest spectral radii among all unicyclic hypergraphs and the first three over all bicyclic hypergraphs are determined, together with two orderings of the corresponding hypergraphs. For topic 2, we investigate the newly introduced odd-colorable hypergraphs and employ their symmetric spectra to obtain conditions for a uniform hypergraph to have equal Laplacian spectrum and signless Laplacian spectrum. For the last topic, some spectral bounds in terms of graph invariants are extended from uniform case to general hypergraphs, and a new way is found to bound the spectral radius from below for a special class of non-uniform hypergraphs. Moreover, the property of symmetric spectrum for general hypergraphs is investigated. Equivalent conditions are extened from uniform case to general case. Besides, the capability of a non-uniform hypergraph to have symmetric (H-)spectrum, equal Laplacian (H-)spectrum (spectral radius) and signless Lapalcian (H-)spectrum (spectral radius) is discussed.