On digraphs determined by their singular values

Abstract: Let $D$ be an digraph of order $n$ with adjacency matrix $A(D)$ and outdegree matrix $Δ^+=Δ^+(D)$. Then the Laplacian and signless Laplacian matrices of $D$ are respectively defined as $L(D)=Δ^+-A(D)$ and $Q(D)=Δ^++A(D)$. In this paper, we compute singular values and an exact formula for the trace norm of Laplacian matrices of the directed path $\overrightarrow{P_n}$, the directed cycle $\overrightarrow{C_n}$ and all orientations of a star. We show that for a bipartite digraph $D$, the matrices $L(D)$ and $Q(D)$ have same singular values and use this to compute the singular values and trace norm of signless Laplacian matrices. We study the problem of determination of digraphs by their singular values and prove the directed path $\overrightarrow{P_n}$, the directed cycle $\overrightarrow{C_n}$ and oriented star $\overrightarrow{S}_n(n-1,0)$ are determined by their Laplacian and signless Laplacian singular values but are not determined by their adjacency singular values.

• The paper establishes that directed paths and cycles are uniquely determined by their Laplacian and signless Laplacian singular values through explicit closed-form expressions.
• It derives trace norms for canonical digraph families, providing precise metrics using sine-based identities for energy computations.
• The study reveals that while most digraphs show unique spectral characteristics, oriented stars are not determined except in the case of all-outgoing arcs from the center.

Digraphs Determined by Their Singular Values: Laplacian and Signless Laplacian Analysis

Introduction and Context

The paper "On digraphs determined by their singular values" (2604.03615) examines the spectral invariants of digraphs through analysis of singular values associated to the Laplacian and signless Laplacian matrices. The work establishes explicit characterizations of these singular values and their associated trace norms for canonical families of oriented graphs, specifically directed paths, cycles, and stars. A central objective is to investigate the problem of whether certain digraphs are uniquely determined (up to isomorphism) by the multiset of their Laplacian (and signless Laplacian) singular values.

This investigation is motivated in part by the broader theory of spectral characterization of graphs, extending key questions from undirected to directed settings and from eigenvalues to singular values, with implications for spectral graph theory, chemistry (through graph energy concepts), and machine learning applications involving digraph spectra.

Definitions and Preliminaries

Let $D$ be a finite simple digraph with adjacency matrix $A(D)$ and diagonal outdegree matrix $\Delta^+$. The Laplacian and signless Laplacian are given by $L(D) = \Delta^+-A(D)$ and $Q(D)=\Delta^++A(D)$, respectively. The spectrum of singular values (the set of positive square roots of the eigenvalues of $LL^T$ or $QQ^T$) plays a role analogous to the classical graph spectrum in spectral characterization studies.

The trace norm (Schatten 1-norm), $\|M\|_*$, is the sum of singular values of matrix $M$, coinciding with the notion of graph energy when $M$ is symmetric.

A digraph $A(D)$0 of order $A(D)$1 is said to be determined by its singular values with respect to a matrix family (e.g., $A(D)$2, $A(D)$3, $A(D)$4) if any $A(D)$5-vertex digraph sharing the same multiset of singular values is isomorphic to $A(D)$6.

Main Results: Explicit Singular Values and Trace Norms

Directed Path and Cycle

For the directed path $A(D)$7 and directed cycle $A(D)$8, the paper presents the following forms for singular values of the Laplacian:

• For $A(D)$9: $\Delta^+$0, $\Delta^+$1.
• For $\Delta^+$2: $\Delta^+$3, $\Delta^+$4.

Closed-form expressions for trace norms of the Laplacian are obtained:

• $\Delta^+$5
• $\Delta^+$6

Both forms are derived via detailed spectral analysis leveraging the symmetric cycle and path adjacency matrix spectra.

Oriented Star Graphs

For the oriented star $\Delta^+$7 (where $\Delta^+$8 arcs are outgoing and $\Delta^+$9 arcs incoming at the center), the singular value distribution is explicitly computed. Most singular values are trivial (0 or 1), except for two, obtained from a quadratic in $L(D) = \Delta^+-A(D)$0. This facilitates direct computation of trace norms for stars and enables analysis of their (non-)uniqueness.

Laplacian and Signless Laplacian Coincidence for Bipartite Digraphs

A structural result proves that for any bipartite digraph $L(D) = \Delta^+-A(D)$1, the matrices $L(D) = \Delta^+-A(D)$2 and $L(D) = \Delta^+-A(D)$3 are unitarily similar and therefore have identical singular value spectra and norms. The proof is elementary and explicit via a simple block-diagonal similarity transformation.

Signless Laplacian Results

Singular values and trace norms for $L(D) = \Delta^+-A(D)$4 and $L(D) = \Delta^+-A(D)$5 are given, showing when they agree with the respective Laplacian norms (precisely when $L(D) = \Delta^+-A(D)$6 is even).

Spectral Characterization by Singular Values

A significant theoretical contribution is in spectral determination:

• Directed path $L(D) = \Delta^+-A(D)$7 and directed cycle $L(D) = \Delta^+-A(D)$8 are both determined by their Laplacian and signless Laplacian singular values. The proof is combinatorial and spectral: for each oriented tree or unicyclic digraph with the same order and possible outdegree sequence, either the sums of squares of singular values (encoded by the first outdegree Zagreb index $L(D) = \Delta^+-A(D)$9 plus the arc count) do not match, or the interlacing property for singular values produces contradictions with the explicit maximal singular value of the canonical digraph.
• These digraphs are not determined by adjacency singular values, as shown by non-isomorphic cospectral examples (e.g., union of a directed cycle and isolated vertex versus a path).
• For oriented stars $Q(D)=\Delta^++A(D)$0, spectral determination by Laplacian or signless Laplacian singular values fails except when $Q(D)=\Delta^++A(D)$1 is $Q(D)=\Delta^++A(D)$2; i.e., for the star where all arcs are outgoing from the center, uniqueness holds. This corresponds to the unique digraph with Laplacian (and signless Laplacian) rank one.

Numerical and Technical Insights

The analytical singular value expressions yield new identities for sine summations, connect trace norms with classical combinatorial invariants, and provide reference formulas for direct computation. Comprehensive inclusion of extremal graph classes rules out all nontrivial non-isomorphic cospectral cases for the canonical path and cycle, providing strong "if and only if" spectral determination statements within the class of all digraphs.

Implications and Future Directions

The results bridge concepts from classical spectral graph theory, matrix analysis, and chemical graph theory (energy), but extend them to the non-symmetric, richer structure of digraphs and singular value domains. For applications in spectral graph isomorphism problems, network identification, and model selection in data science, the explicit criteria for uniqueness may inform the design of graph invariants used for classification and clustering in directed networks.

Future research may address:

• Extension to broader families of digraphs such as regular, Eulerian, or strongly connected classes.
• Characterization of families where neither adjacency nor Laplacian singular values are sufficient for determination, motivating higher-order invariants.
• Algorithmic aspects: efficient recognition and isomorphism testing via singular value data.
• Robustness under noise or perturbations, relevant to applications in dynamic networks.

Conclusion

This paper establishes explicit formulae for Laplacian and signless Laplacian singular values and trace norms for fundamental digraph families, and rigorously characterizes conditions for unique spectral determination by singular values. Directed paths, cycles, and the maximally-outdegree star are unambiguously identified by their Laplacian and signless Laplacian singular value spectra, in sharp contrast to adjacency singular values. The findings reinforce the utility of Laplacian-based singular value spectra in the combinatorial and algebraic analysis of digraphs, and motivate further studies into their spectral uniqueness properties.