Eigenvalues of Hypergraph Products and Reciprocal Eigenvalue Property

Abstract: Spectral hypergraph theory has recently attracted considerable interest as it provides a natural framework for modeling higher-order relationships beyond classical graphs. In this setting, eigenvalues of adjacency, Laplacian, and signless-Laplacian hypermatrices play an important role in understanding the underlying structure of hypergraphs. In this work, we study the adjacency, Laplacian, and signless-Laplacian eigenvalues of the join, Kronecker, and corona products of hypergraphs, and examine how these spectra behave under such operations. These investigations help in better understanding the interplay between hypergraph structure and spectral properties. The reciprocal eigenvalue property is of particular interest due to the spectral symmetries it reflects. Motivated by this, we extend the notion of reciprocal eigenvalue property to hypergraphs and show that power hypertrees do not satisfy this property.

• The paper establishes explicit combinatorial and algebraic characterizations for the spectral behavior of hypergraph join, Kronecker, and corona products.
• It introduces a multiplicative framework for Kronecker products and a recursive polynomial structure for corona products, linking eigenvalue constructions to characteristic equations.
• The paper reveals that the reciprocal eigenvalue property fails for general r-uniform hypertrees, highlighting inherent algebraic asymmetries in higher-order networks.

Spectral Analysis of Hypergraph Products and the Reciprocal Eigenvalue Property

Introduction

The study of spectral properties of hypergraphs, fueled by their capacity to encode higher-order interactions absent in classical graphs, has drawn significant mathematical and algorithmic interest. Extending established results on the spectra of graph products, this paper systematically investigates how adjacency, Laplacian, and signless Laplacian spectra behave under hypergraph join, Kronecker, and corona products. A particular focus is on the structure and invariance of eigenvalues under these operations, and, notably, on the generalization and limitations of the reciprocal eigenvalue property (REP) within hypergraphs—especially in the context of power hypertrees.

Hypermatrices and Hypergraph Spectra

The extension of matrix theory to tensors (hypermatrix framework) enables a formal approach to eigenvalues and eigenvectors for uniform hypergraphs. For an $r$-uniform hypergraph, the adjacency, Laplacian, and signless-Laplacian are represented as symmetric $r$-order hypermatrices whose spectra encapsulate structural features of the hypergraph.

The eigenvalue problem for an order $r$ hypermatrix $\mathcal{T}$ is defined by the system $$\mathcal{T}\mathbf{y}^{r-1} = \lambda\mathbf{y}^{[r-1]}$$, generalizing the matrix ($r=2$) scenario. The concept of main and non-main eigenvalues is developed in analogy to the graph case, allowing a refined analysis of spectral interlacing and multiplicity phenomena.

Spectra of Hypergraph Products

Join of Hypergraphs

Explicit combinatorial and algebraic characterizations are derived for the spectra of join products. If $\mathcal{H}_1$ and $\mathcal{H}_2$ are $r$-uniform with orders $n_1$ and $r$0, then the non-main eigenvalues of the Laplacian of the join $r$1 include the non-main eigenvalues of the factors. The remaining eigenvalues are determined by roots of parametric polynomial equations dependent on $r$2, $r$3, and the uniformity $r$4, typically of degree $r$5. Analogous results are provided for adjacency and signless-Laplacian spectra, with spectral radii associated to real positive solutions.

Kronecker Product

For Kronecker products $r$6, the adjacency spectrum admits a multiplicative construction: if $r$7 and $r$8 are eigenpairs for $r$9, $r$0 respectively, then $r$1 is an eigenpair for the product. For regular hypergraphs, Laplacian eigenvalues decompose as $r$2, with analogous formulas for the signless-Laplacian.

Corona Product

The analysis reveals that for corona products, the spectra exhibit a recursive structure tied to the spectra of the factors and roots of characteristic polynomials reflecting iterative pendant attachment. The adjacency spectrum of the corona with $r$3-cliques, for instance, is coded by a degree $r$4 polynomial $r$5. The Laplacian and signless-Laplacian spectra follow similar polynomial transformations, with explicit multiplicities.

Reciprocal Eigenvalue Property in Hypergraphs

The classical REP and its strong variant (SRP) in graphs—where the spectrum is symmetric under inversion—are reconsidered in the hypergraph context. The generalization faces fundamental obstructions: for $r$6-uniform hypertrees with $r$7, the presence of zero as an omnipresent eigenvalue and the combinatorial structure of the characteristic polynomial preclude the spectrum (including the nonzero part) from being closed under inversion, i.e., the spectrum is not palindromic except in degenerate situations. This is evidenced by characteristic polynomials of the form $r$8 occurring as factors, which preclude the REP unless $r$9.

A combinatorial algebraic approach connects the matching polynomial of the hypertree to the palindromicity of the characteristic polynomial, establishing that (SRP) holds if and only if all sub-hypertree matching polynomials are palindromic—a rare scenario for general $\mathcal{T}$0-uniform hypertrees.

Moreover, maximal subsets of the spectrum obeying inverse closure are characterized, and it is shown that only hyperstar and small path hypergraphs (e.g., $\mathcal{T}$1, $\mathcal{T}$2) attain substantial symmetry in this sense.

Theoretical Implications and Prospects

The results provide a rigorous foundation for extending the spectral graph product machinery into the hypergraph domain, clarifying the transferability and limitations of product-based spectral symmetries. The precise spectral decompositions enable refined analysis of higher-order network models, with implications for spectral clustering, diffusion processes, and higher-order random walks on hypergraphs.

The failure of REP for general $\mathcal{T}$3-uniform hypertrees highlights deep algebraic asymmetries intrinsic to high-order combinatorics, posing challenges for extending spectral invariants and duality phenomena observed in graphs to true hypergraph regime. This negative result suggests that spectral algorithms or isomorphism invariants reliant on REP must be carefully adapted or replaced when dealing with non-graphical uniform hypergraphs.

There is also a clear pathway for future advancements: classification of those restricted hypergraph families whose matching polynomials, or related invariants, admit palindromicity, as well as exploration of spectral symmetries in signed or weighted hypergraph settings. Analytical tools connecting the spectra of hypergraph products to combinatorial properties, such as matching and treewidth, are expected to further mature.

Conclusion

This work delivers explicit, algebraic descriptions for how spectra of adjacency, Laplacian, and signless-Laplacian tensors behave under core hypergraph operations: join, Kronecker, and corona products. It establishes that the familiar reciprocal eigenvalue property does not generalize to $\mathcal{T}$4-uniform hypertrees for $\mathcal{T}$5, except in trivial cases. These findings deepen our theoretical understanding of hypergraph spectra and delineate the boundaries of graph-derived spectral invariants, guiding both theoretical advances and algorithmic applications in higher-order network science.

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Reference: "Eigenvalues of Hypergraph Products and Reciprocal Eigenvalue Property" (2604.23365)