Characteristic adjacency matrix associated with a hypergraph

Abstract: We define a new matrix associated with a hypergraph. A study of this matrix is carried out, in particular the study of its spectrum, which shows the relevance of the construction. We demon- strate that this adjacency matrix is characterisitic of the hypergraph, that is to say, two isomorphic hypergraphs have similar matrices. Furthermore, starting from this matrix, we can reconstruct the hypergraph. Starting from this matrix, we introduce new graphs associated with the hypergraph.

• The paper introduces a Gaussian-integer-valued Hermitian matrix indexed by vertex–hyperedge incidences, proving that it characterizes hypergraphs up to isomorphism without uniformity or linearity assumptions.
• The matrix has real, order-independent eigenvalues with bounds linked to hypergraph rank and maximum degree, interlacing for subhypergraphs, trace identities, and a positive semidefinite Laplacian.
• The associated graph refines both the line graph and 2-section through quotients, while the polynomial-time Hermitian framework offers a practical alternative to computationally hard tensor-based spectral methods.

Motivation: the failure of classical adjacency matrices for hypergraphs

In algebraic graph theory, the adjacency matrix is a complete invariant of a graph up to isomorphism, and virtually all spectral tools (Laplacians, characteristic polynomials, interlacing) derive from it. For hypergraphs the situation is fundamentally different: the usual incidence-based adjacency matrix is not characteristic of the hypergraph, so two non-isomorphic hypergraphs can share the same matrix data. The dominant remedy in the literature has been to encode hypergraphs as tensors or hypermatrices [Cooper2012, Hou2019, Liu2016, Cardoso2019], but this comes at a steep computational price: eigenvalue computation and most tensor problems are NP-hard [Hillar2013, BANERJEE2021], which severely limits algorithmic exploitation of tensor spectral results.

The paper by Bretto and Faisant proposes an alternative: a new representation of a hypergraph $H=(V;E,\varepsilon)$ as the incidence set

$$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$

and an associated Hermitian adjacency matrix whose rows and columns are indexed by these pairs $(x,e)$ rather than by vertices or edges alone. The construction is deliberately designed so that the matrix determines the hypergraph uniquely — addressing the characterization defect directly while remaining an ordinary (Hermitian) matrix, hence avoiding the NP-hardness of the tensor approach.

Definition of the endomorphism $\mathfrak{A}$ and the matrix $A_H$

Fixing orders on $V$ and $E$, the set $VE$ is viewed as an orthonormal basis of a Hermitian space of dimension $m_E = \sum_{e\in E}|\varepsilon(e)|$. The endomorphism $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$ is defined on basis vectors by

$$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$0

where $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$1 denotes the imaginary unit. In matrix form, entries are Gaussian integers: entry $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$2 between distinct vertices sharing a hyperedge, and $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$3 between pairs sharing a vertex across different hyperedges, with sign determined by the chosen order on $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$4. Under the reverse lexicographic ordering ($$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$5), each hyperedge $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$6 yields a maximal principal "1-block" of size $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$7; under lexicographic ordering ($$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$8), each vertex $$H := V \ast E = \{(x,e) \in V\times E : x\in\varepsilon(e)\},$$9 yields a maximal principal "$(x,e)$0-block" of size $(x,e)$1, skew-Hermitian in structure.

Two structural facts follow immediately. First, $(x,e)$2 is Hermitian, hence diagonalizable with real spectrum — a decisive practical advantage over tensors. Second, changing the orders on $(x,e)$3 and $(x,e)$4 only conjugates the endomorphism, producing similar matrices, so the spectral content is order-independent. A duality rule is also established: the matrix of the dual hypergraph $(x,e)$5 is obtained from $(x,e)$6 by replacing $(x,e)$7 with $(x,e)$8 and converting off-diagonal $(x,e)$9s into ordered $\mathfrak{A}$0 entries.

Spectral properties

The paper develops a Rayleigh-quotient analysis yielding quantitative bounds. The main results are:

Property	Statement
Spectral bound	$\mathfrak{A}$1
Extremal values	If some $\mathfrak{A}$2 has $\mathfrak{A}$3, then $\mathfrak{A}$4
Trace identities	$\mathfrak{A}$5; $\mathfrak{A}$6
Parity	$\mathfrak{A}$7 is even
Interlacing	Eigenvalues of induced subhypergraphs and partial hypergraphs interlace those of $\mathfrak{A}$8

For $\mathfrak{A}$9-uniform hypergraphs, the bound specializes to $A_H$0, connecting the new spectrum to the 2-section. Using interlacing against the principal submatrix $A_H$1 (the Hermitian matrix with $A_H$2 above the diagonal, analyzed via arguments inspired by Guo–Mohar's work on Hermitian adjacency matrices of mixed graphs [Guo2015]), the authors prove that $A_H$3 (and $A_H$4 when $A_H$5 is odd), and that the spectrum contains at least $A_H$6 positive and $A_H$7 negative eigenvalues. These results show that the spectrum carries genuine combinatorial information about degrees, rank, and maximum degree — supporting the claim that the construction is spectrally meaningful, not merely a bookkeeping device.

A Laplacian $A_H$8, with diagonal degree $A_H$9, is introduced and shown to be positive semidefinite via a diagonal-dominance argument, with trace equal to $V$0.

Characterization theorem

The central result is that $V$1 fully characterizes the hypergraph. Specifically, $V$2 if and only if there exists a linear isomorphism $V$3 mapping $V$4 bijectively onto $V$5 such that $V$6. The forward direction follows from functoriality of the construction under isomorphisms; the converse is proved by extracting from any such intertwining isomorphism bijections $V$7 and $V$8 satisfying $V$9, using the block structure of the matrix to identify vertex-pairs within hyperedges and edge-pairs through vertices. This is the property that classical hypergraph adjacency matrices lack, and it holds without any uniformity or linearity assumptions. It also implies that the associated mixed digraph interpretation of $E$0 characterizes the hypergraph up to isomorphism.

Associated graphs

From $E$1 the authors define the associated graph $E$2, joining $E$3 and $E$4 whenever the corresponding matrix entry is nonzero, and interpret $E$5 itself as the adjacency structure of a mixed graph $E$6 (undirected edges for the $E$7-entries, directed arcs weighted $E$8 for the $E$9-entries). Two quotient constructions recover classical objects:

• Quotienting $VE$0 by the relation $VE$1 yields exactly the line graph $VE$2.
• Quotienting by $VE$3 yields exactly the 2-section $VE$4.

Thus $VE$5 simultaneously refines both standard graph approximations of a hypergraph. Additionally, for simple linear hypergraphs with minimum degree at least $VE$6, the associated graph satisfies $VE$7, i.e., it is self-dual. Decomposing the absolute-value adjacency matrix $VE$8 into real and imaginary parts $VE$9, one obtains $m_E = \sum_{e\in E}|\varepsilon(e)|$0 as an eigenvalue of $m_E = \sum_{e\in E}|\varepsilon(e)|$1 when $m_E = \sum_{e\in E}|\varepsilon(e)|$2 is $m_E = \sum_{e\in E}|\varepsilon(e)|$3-uniform and of $m_E = \sum_{e\in E}|\varepsilon(e)|$4 when $m_E = \sum_{e\in E}|\varepsilon(e)|$5 is $m_E = \sum_{e\in E}|\varepsilon(e)|$6-regular.

Limitations and open questions

Several caveats should be noted. The matrix dimension is $m_E = \sum_{e\in E}|\varepsilon(e)|$7, which can substantially exceed both $m_E = \sum_{e\in E}|\varepsilon(e)|$8 and $m_E = \sum_{e\in E}|\varepsilon(e)|$9 for dense hypergraphs, so the computational advantage over tensors rests on polynomial-time Hermitian eigensolvers rather than on small matrix size. The construction requires excluding isolated vertices and empty hyperedges for the reconstruction argument, and loops force $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$0 into the spectrum, slightly complicating extremal statements. The characterization theorem requires the intertwining map to restrict to a bijection on $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$1; whether this hypothesis can be weakened is left unaddressed. The authors explicitly pose open problems: which mixed graphs arise as $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$2 for some hypergraph (and similarly for $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$3), whether an efficient reconstruction algorithm from $\mathfrak{A}\in\mathrm{End}(\langle VE\rangle)$4 can be developed, and whether the spectrum informs hypergraph colorings. These questions remain unanswered in the paper.

Conclusion

This paper introduces a Hermitian, Gaussian-integer-valued adjacency matrix indexed by vertex–hyperedge incidences, proves that it characterizes the hypergraph up to isomorphism — resolving the principal deficiency of classical hypergraph matrices — and establishes a coherent spectral theory including eigenvalue bounds tied to rank and maximum degree, interlacing, a positive semidefinite Laplacian, and quotients recovering the line graph and 2-section. By staying within ordinary Hermitian linear algebra, the construction sidesteps the NP-hardness inherent in tensor approaches, though questions of graph realizability, reconstruction algorithms, and chromatic applications remain open.