Abstract
The p-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex p-Laplacian and a hyperedge p-Laplacian are defined for oriented hypergraphs, for all p ≥ 1. Several spectral properties of these operators are investigated.
Highlights
Oriented hypergraphs are hypergraphs with the additional structure that each vertex in a hyperedge is either an input, an output or both
The p-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. Both a vertex p-Laplacian and a hyperedge p-Laplacian are defined for oriented hypergraphs, for all p ≥ 1
While the vertex p-Laplacian is a known operator for graphs, to the best of our knowledge the only edge p-Laplacian for graphs that has been defined is the classical one for p = 2
Summary
Oriented hypergraphs are hypergraphs with the additional structure that each vertex in a hyperedge is either an input, an output or both. They have been introduced in [21], together with two normalized Laplace operators whose spectral properties and possible applications have been investigated in further works [1, 31,32,33]. We generalize the Laplace operators on oriented hypergraphs by introducing, for each p ∈ R≥1, two p-Laplacians. Related Work It is worth mentioning that, in [18], other vertex p-Laplacians for hypergraphs have been introduced and studied. [18] focuses on classical hypergraphs, while we consider, more generally, oriented hypergraphs
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