Characterizing uniform hypergraphs via Seidel matrix and Seidel energy

Abstract: The Seidel energy is defined as the sum of the absolute values of the eigenvalues of the Seidel matrix of a hypergraph. We first characterize the k-uniform hypergraphs of fixed order n with minimum and maximum Frobenius norms of Seidel matrices and then derive bounds for the Seidel energy. Building on these results, we obtain a negative answer to the hypergraph analogue of Haemers Conjecture by showing that the complete k-uniform hypergraph does not, in general, minimize Seidel energy. Motivated by the theory of hypoenergetic and non-hypoenergetic graphs, we define Seidel hypoenergetic and Seidel non-hypoenergetic hypergraphs and prove that almost all k-uniform hypergraphs are Seidel non-hypoenergetic.

• The paper introduces a generalized Seidel matrix for k-uniform hypergraphs and defines its associated spectral energy.
• It disproves Haemers’ Conjecture by showing that, unlike graphs, complete hypergraphs do not always minimize Seidel energy.
• Probabilistic analysis reveals that almost all large k-uniform hypergraphs are non-hypoenergetic, establishing new bounds on spectral extremals.

Characterizing Uniform Hypergraphs via Seidel Matrix and Seidel Energy

Introduction and Motivation

This work systematically analyzes the Seidel matrix and Seidel energy for $k$-uniform hypergraphs, extending foundational concepts from spectral graph theory. The Seidel matrix, traditionally a central object for graphs in relation to switching theory and extremal spectral problems, is non-trivial to extend to hypergraphs due to the multi-way nature of hyperedges, which induce higher-order dependencies in co-degrees and spectral structure. The paper formalizes the construction of the Seidel matrix for $k$-uniform hypergraphs, develops extremal results for its Frobenius norm and spectral energy, disproves the validity of Haemers’ Conjecture in the hypergraph regime, and addresses the probabilistic prevalence of Seidel non-hypoenergeticity in the Erdős–Rényi-inspired random hypergraph model.

Seidel Matrix and Energy: Generalization and Extremal Norms

Let $\mathcal{H}$ be a $k$-uniform hypergraph on $n$ vertices. The Seidel matrix is defined as

$$\mathcal{S}(\mathcal{H}) = J_n - I_n - 2\mathcal{A}(\mathcal{H})$$

where $\mathcal{A}(\mathcal{H})$ is the canonical $n\times n$ adjacency matrix for $\mathcal{H}$, and the off-diagonal entry $\mathcal{S}(\mathcal{H})_{ij} = 1-2c_{ij}$, with $k$0 the co-degree (number of hyperedges containing both $k$1). The Seidel energy $k$2 is the sum of the absolute values of the eigenvalues of $k$3.

The Frobenius norm of $k$4 is explicitly related to hypergraph co-degrees: $k$5 A critical structural dichotomy is established:

• The linear $k$6-uniform hypergraphs (where every pair of vertices is contained in at most one hyperedge) uniquely attain the minimal norm $k$7.
• Complete $k$8-uniform hypergraphs ($k$9) maximize the Frobenius norm, with explicit value $\mathcal{H}$0.

The interrelation between the Frobenius norm and the Seidel energy is tightly controlled by classical norm inequalities, enforcing

$\mathcal{H}$1

This leverages classical convexity arguments and permits transfer between norm extremal results and energy bounds.

Failure of Haemers’ Conjecture in the Hypergraph Regime

Haemers’ Conjecture asserts that, for graphs of order $\mathcal{H}$2, the complete graph minimizes Seidel energy. This property, validated in several works for various classes of graphs, is shown in this paper to fail in the context of $\mathcal{H}$3-uniform hypergraphs with $\mathcal{H}$4.

Using explicit computation, the Seidel energy of the complete $\mathcal{H}$5-uniform hypergraph on $\mathcal{H}$6 vertices is

$\mathcal{H}$7

However, specific sparse $\mathcal{H}$8-uniform hypergraphs (notably, the hypertriangle $\mathcal{H}$9 on $k$0 vertices, consisting of three hyperedges with specified overlap) can be constructed so that

$k$1

This is substantiated by both closed-form eigenvalue calculations for $k$2 and explicit asymptotic estimation for arbitrarily large $k$3. Thus, the natural hypergraph analogue of Haemers’ Conjecture is definitively false in all cases $k$4.

Seidel Hypoenergeticity in Random Uniform Hypergraphs

The concept of (classical) hypoenergetic and non-hypoenergetic graphs is extended to the Seidel energy for uniform hypergraphs. Employing the random hypergraph model $k$5, each possible $k$6-hyperedge is present independently with probability $k$7. For fixed $k$8 and $k$9,

$n$0

and, via Chernoff bounds, $n$1 concentrates sharply around its mean (with high probability across all vertex pairs as $n$2).

Asymptotically, for almost all random $n$3-uniform hypergraphs,

$n$4

which dominates the linear threshold $n$5 for all $n$6. Thus, almost all $n$7-uniform hypergraphs are Seidel non-hypoenergetic.

Structured Partitions and Seidel Quotients

Recognizing that direct spectral analysis is generally intractable for hypergraphs, the framework adapts the concept of equitable partitions to the Seidel matrix of hypergraphs. This allows the reduction of spectral calculations to those of quotient matrices when the hypergraph admits a Seidel-equitable partition, paralleling similar reductions in the Laplacian and adjacency matrix settings. For instance, in the study of hypertriangles, the Seidel spectrum computation is facilitated by reducing to a quotient matrix, whose eigenvalues are contained in the original spectrum.

Implications and Future Work

The results concretely demonstrate that spectral extremal phenomena for graphs—such as minimization of energy by the complete graph—do not necessarily generalize to the hypergraph setting when using Seidel-type constructions. This failure underscores the increased combinatorial and spectral complexity inherent in uniform hypergraphs and motivates the investigation of new structural types beyond completeness for spectral extremality.

The probabilistic results indicate that Seidel non-hypoenergeticity is overwhelmingly typical for large uniform hypergraphs. This sets a baseline for understanding energetic and spectral behaviors to be expected in random hypergraph models, which potentially impacts theoretical approaches to random processes on hypergraphs and the design of algorithms exploiting such spectral invariants.

Several open problems are formulated:

• Explicit characterization of Seidel hyperenergetic and non-hyperenergetic $n$8-uniform hypergraphs.
• Description of extremal uniform hypergraphs (maximizers and minimizers) for Seidel energy and Frobenius norm in both unrestricted and regular cases.

Conclusion

This work provides a comprehensive analysis of the Seidel matrix and Seidel energy for $n$9-uniform hypergraphs, yielding sharp extremal bounds, counterexamples to direct generalizations of graph spectral conjectures, novel applications of probabilistic concentration, and structural tools for spectral analysis. These results clarify the differences between graph and hypergraph spectral theory and open new avenues for the investigation of spectral extremals, randomness, and structure in hypergraphs.

Reference: "Characterizing uniform hypergraphs via Seidel matrix and Seidel energy" (2606.17817)