On the spectrum of hypergraphs

• Published in: Linear algebra and its applications Vol. 614; pp. 82 - 110
• Main Author: Banerjee, Anirban
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.04.2021; American Elsevier Company, Inc
• Subjects: Cheeger constant; Eigenvalues; General hypergraph; Graph theory; Graphs; Linear algebra; Random walk; Random walk on hypergraphs; Ricci curvature of hypergraphs; Spectral theory of hypergraphs; Transition probabilities; 05C65; General hypergraph; 05C50; Cheeger constant; 05C40; 47A75; Ricci curvature of hypergraphs; Spectral theory of hypergraphs; 05C15; 15A18; Random walk on hypergraphs
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2020.01.012

Here, we introduce different connectivity matrices and study their eigenvalues to explore various structural properties of a general hypergraph. We investigate how the diameter, connectivity and vertex chromatic number of a hypergraph are related to the spectrum of these matrices. Different properties of a regular hypergraph are also characterized by the same. Cheeger constant on a hypergraph is defined and its spectral bounds have been derived for a connected general hypergraph. Random walk on a general hypergraph can also be well studied by analyzing the spectrum of the transition probability operator defined on the hypergraph. We also introduce Ricci curvature on a general hypergraph and study its relation with the hypergraph spectra.