On Some General Operators of Hypergraphs

TL;DR

This paper introduces a unified framework of connectivity operators for hypergraphs, generalizing traditional matrices, and explores their spectral properties and applications in networks, disease modeling, and combinatorial bounds.

Contribution

It presents a novel unified framework for hypergraph connectivity operators, extending classical notions and enabling new spectral and application analyses.

Findings

Eigenvalues and eigenspaces computed for certain hypergraph classes.

Spectral bounds derived for hypergraph invariants.

Applications demonstrated in random walks, network dynamics, and disease spread.

Abstract

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different connectivity matrices associated with hypergraphs. In fact, we introduce here a unified framework for studying different variations of the connectivity operators associated with hypergraphs at the same time. Eigenvalues and corresponding eigenspaces of the general connectivity operators associated with some classes of hypergraphs are computed. Applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are studied from the perspective of our newly introduced operators. We also derive spectral bounds for the weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant of hypergraphs.