One can almost never hear the shape of a digraph

Abstract: Kac raised the question `Can one hear the shape of a drum?' The answer is negative. On the other hand, one can hear the shape of a drum in the generic case. Haemers conjectured that almost all graphs are determined by their spectra, which can be regarded as a discrete version of Kac's question. Vu conjectured a similar phenomenon for symmetric $\pm 1$ matrices. The first main result of this paper shows that one can almost never hear the shape of a digraph by proving that almost all digraphs are not isomorphic to their reverse and almost all digraphs have trivial automorphism groups. This indicates that the real symmetric condition is vital in both Haemers' conjecture and Vu's conjecture. The second main result of this paper shows that almost all graphs of order $n$ have no cospectral mates with height $o(( n / \ln n)^{1/10})$, improving an earlier result on cospectral mates with fixed level.

• The paper demonstrates that almost all digraphs have non-isomorphic cospectral mates, showing the spectrum rarely determines a digraph uniquely.
• It employs probabilistic and combinatorial methods to reveal that reverse digraphs share identical spectra yet are typically non-isomorphic.
• For undirected graphs, improved bounds on conjugator heights reinforce strong spectral distinctions and highlight the role of symmetry in determination.

Spectral Determination of Digraphs and Cospectral Mates: Asymptotic Impossibility

Introduction and Context

The paper "One can almost never hear the shape of a digraph" (2604.02165) investigates to what extent the spectrum of the adjacency or Hermitian adjacency matrix determines a finite digraph (directed graph) up to isomorphism. This question is a directed graph analogue of Kac's celebrated problem—"Can one hear the shape of a drum?"—and Haemers’ conjecture for undirected (simple) graphs, which asserts that almost all undirected graphs are determined by their spectra. The problem is conceptually tied to foundational issues in spectral graph theory, random matrix theory, and combinatorial isomorphism testing.

Main Results

Asymptotic Non-Determination of Digraphs by Spectrum

The principal result asserts that for a uniformly random (Erdős–Rényi-type) digraph $D\sim D(n,p)$, the probability that $D$ is determined (up to isomorphism) by its adjacency or Hermitian adjacency spectrum converges to zero as $n\to\infty$. That is:

• Almost all digraphs possess at least one nontrivial cospectral mate; generically, the spectrum is insufficient for isomorphism determination.
• The underlying mechanism is that the reverse digraph $D^{R}$—obtained by reversing all arc directions—shares the same spectrum as $D$, yet, with high probability, $D$ is not isomorphic to $D^{R}$. This sharply diverges from the situation for undirected graphs, where such dualities and generic indistinguishability are far rarer.

Furthermore, it is established that almost all digraphs have trivial automorphism groups, generalizing classical results for undirected graphs. The proof uses growth rate estimates for the probability that any non-identity permutation acts as an automorphism, exploiting independence across edges and orbit structure under the permutation group action on ordered pairs.

Improved Bounds for Cospectral Mates in the Graph Case

In the undirected graph case, the work significantly strengthens previous results on cospectral mates with rational orthogonal conjugating matrices. Let the height of such a matrix be the maximum denominator among its entries. The results are:

• For each fixed integer $h\geq 2$, almost all graphs of order $n$ have no cospectral mate with a conjugator of height at most $h$.
• Moreover, this nonexistence result can be extended to the case where $D$0, covering a broad range of possible conjugating matrices as $D$1.

These results generalize and sharpen those of Wang and Zhao (Wang et al., 6 Sep 2025), transitioning from controls based on the level (the least common denominator) to the more robust notion of height, and removing the necessity for summation bounds over rational levels.

Theoretical and Practical Implications

The results emphasize the critical role of symmetry structure in spectral determination:

• For undirected graphs (real symmetric adjacency matrices), almost all graphs are believed to be determined by spectra (Haemers’ conjecture), whereas for digraphs (nonsymmetric adjacency matrices), almost all digraphs are not. This demonstrates that the real symmetric condition is not a technicality but a decisive factor differentiating these combinatorial universes.
• The results have implications for graph isomorphism heuristics based on spectral invariants, indicating such approaches will almost never resolve isomorphism for large random digraphs.

Additionally, the bounds derived for heights of rational conjugators in the undirected case suggest strong spectral distinction among generic graphs, hinting that cospectrality is strictly limited outside exceptional, highly structured cases.

Analytical Approach

The probabilistic proofs rely crucially on independence properties of the random digraph model. For automorphism-related statements, union bounds combined with careful enumeration of permutation effects and orbit analysis suffice to show vanishing probabilities for nontrivial symmetries in the large-$D$2 regime. For conjugator height analysis, the approach utilizes canonical reduction for rational orthogonal matrices and estimates the number and interaction of such conjugators with the adjacency matrices under randomness, leveraging combinatorial and probabilistic concentration.

Relationship to Prior and Ongoing Work

The paper contextualizes its results within the substantial literature on the spectral characterization problem for graphs, referencing canonical works by van Dam and Haemers and ongoing developments, such as the recent introduction of multivariate and block diagonal graph spectra (Xiang, 31 Jan 2026, Xu et al., 28 Nov 2025), which can circumvent the negative results for standard spectra. These approaches offer potential for universal invariants beyond the adjacency/Laplacian paradigm, notably for real symmetric matrices.

Future Directions

Natural research directions arising from this work include:

• Explicit construction and efficient computation of spectral invariants (functions $D$3) with maximal discriminative power over graph isomorphism classes.
• Determination of whether there exists a fixed or ‘natural’ function $D$4 (independent of the graph order) such that the spectrum of $D$5 asymptotically determines all graphs.
• Extension of the present methods to weighted digraphs, multigraphs, or hypergraphs, where symmetry constraints and spectra structure are more complex.

The results invite further investigation into spectral approaches for graph isomorphism, especially where the underlying matrices are not real symmetric but retain enough structure for spectral descriptors to be potentially informative.

Conclusion

This work rigorously demonstrates that the spectrum almost never determines the shape (isomorphism class) of a random digraph, in stark contrast to the conjectural situation for undirected graphs. This highlights a fundamental limitation in the employment of spectral invariants for graph isomorphism in the directed setting and underscores the necessity of real symmetric matrix conditions in achieving strong spectral characterizations. The improvements in bounds for cospectral mates in the undirected case further clarify the landscape, setting stringent asymptotic barriers for the existence of cospectral but non-isomorphic random graphs with bounded height conjugators. The theoretical framework provided here sets a baseline for further exploration in spectral graph theory and its algorithmic ramifications.