Incidence-adjacency-Laplacian relations for positive-weight directed hypergraphs

Determine the relationship among incidence matrices, adjacency matrices, and Laplacians for directed hypergraphs with positive weights.

Background

The manuscript notes that standard relationships among incidence matrices, adjacency matrices, and Laplacians are known in some graph and directed-graph settings but become less straightforward for directed hypergraphs. In particular, the text highlights ambiguity in directed incidence conventions and the possible need for separate in-incidence and out-incidence matrices.

References

What is the relation between incidence matrix ($\mathcal{I}$), adjacency matrix ($A$) and Laplacian $\mathcal{L}$ in the theory of dihypergraphs (with positive weights).

— Pangraphs as models of higher-order interactions  (2502.10141 - IskrzyĹ„ski et al., 14 Feb 2025) in Section 'Additional copy', Subsubsection 'Weighted multilayer dihypergraph'