Signed & Unsigned Adjacency Matrices

• Signed and unsigned adjacency matrices represent a graph's edge structure, with unsigned matrices indicating simple connections and signed matrices incorporating ±1 to denote edge polarity.
• They extend classical spectral graph theory by enabling analysis of directed, weighted, and complex graphs through adaptations like complex Hermitian and effective adjacency formulations.
• Their spectral properties, including balance, switching equivalence, and perturbation effects, drive practical insights in clustering, graph signal processing, and machine learning.

A signed or unsigned adjacency matrix encodes the edge structure of a graph; its specific construction depends on whether the edges are undirected/directed and whether they possess associated signs (typically ±1). In unsigned graphs, the adjacency matrix forms the backbone of classical spectral graph theory, clustering, and network analysis. When edges are allowed to carry negative signs, or to be directed, a richer array of combinatorial and spectral phenomena emerges, leading to generalized matrix constructions and new theoretical challenges. Recent research has further extended the formalism to signed directed graphs and to effective adjacency matrices arising from deformed Laplacians, enabling advanced graph signal processing and machine learning methods.

1. Fundamental Definitions: Unsigned and Signed Adjacency Matrices

Let $G = (V, E)$ be a finite, simple graph with $n=|V|$ vertices.

• Unsigned adjacency matrix: For an undirected, unsigned graph,

$$A(G)_{ij} = \begin{cases} 1 & \text{if } \{v_i,v_j\} \in E, \ 0 & \text{otherwise}. \end{cases}$$

$A(G)$ is symmetric and has zero diagonal for simple graphs. For directed unsigned graphs, the matrix is generally asymmetric:

$$A_{ij} = \begin{cases} 1 & \text{if there is an edge } i \to j, \ 0 & \text{otherwise}. \end{cases}$$

• Signed adjacency matrix: For a signed simple graph $\Sigma = (G, \sigma)$, with signature assignment $\sigma: E \to \{+1, -1\}$,

$$A^\Sigma_{ij} = \begin{cases} \sigma(ij) & \text{if } \{v_i,v_j\} \in E, \ 0 & \text{otherwise}. \end{cases}$$

The matrix is symmetric for undirected signed graphs and encodes both the connectivity and the sign structure (Zaslavsky, 2013, Belardo et al., 2019, Li et al., 2022).

• Generalizations: For edge weights in $\mathbb{R}$ (possibly negative), or for directed, signed, weighted graphs, $A_{ij} = w(i,j)$ if $(i \to j) \in E$, $0$ otherwise, without symmetry in the general case (Resende, 2024).

2. Extensions to Signed Directed Graphs and Complex Hermitian Adjacent Matrices

The classical approaches break Hermitian symmetry for signed directed graphs, necessitating new constructions:

• The complex Hermitian adjacency matrix encodes simultaneously the presence, sign, and direction of edges. This is constructed as follows (Ko et al., 2022):
  1. Define $A$ as the ordinary directed adjacency matrix, and set $A_s = (A + A^T)/2$ (the undirected connectivity, with entries in $\{0, \frac12, 1\}$).
  2. Define a phase matrix $P^q \in \mathbb{C}^{n \times n}$:

  $$P^q(u,v) = \frac{e^{i\Theta^q(u,v)}A_{uv} + e^{i\overline{\Theta}^q(u,v)}A_{vu}}{\|e^{i\Theta^q(u,v)}A_{uv} + e^{i\overline{\Theta}^q(u,v)}A_{vu}\| + \epsilon}$$

  with angles $\Theta^q(u,v)$ and $\overline{\Theta}^q(u,v)$ determined by the sign and direction of the edge. 3. The Hermitian adjacency is

  $H^q = A_s \odot P^q$

  using the Hadamard (elementwise) product.

• The modulus encodes the "multiplicity" of bidirectionality; the argument (phase) encodes sign ($\pi$ phase shift for negative) and orientation ($\pm q$ offset for direction).

This formulation yields a Hermitian matrix for arbitrary signed directed graphs, preserving positive semidefiniteness of Laplacians and supporting spectral graph methods beyond unsigned or undirected cases (Ko et al., 2022).

3. Spectral Properties, Balance, and Switching

Key spectral phenomena for signed matrices include (Zaslavsky, 2013, Belardo et al., 2019, Li et al., 2022, Kannan et al., 2022):

• Switching equivalence: For any $\eta: V \to \{\pm1\}$, switching the signature via $\sigma'(ij) = \eta(i)^{-1}\sigma(ij)\eta(j)$ gives $A(\Sigma') = D_\eta^{-1} A(\Sigma) D_\eta$. Thus, spectrum is invariant under switching.
• Balance: A cycle is balanced if the product of its edge signs is $+1$. $\Sigma$ is balanced iff $A^\Sigma$ is similar to $A(G)$; spectra coincide (Acharya's theorem).
• Spectral properties:
  • The eigenvalues of $A^\Sigma$ are bounded above by those of the underlying $A(G)$.
  • For $k$-regular $G$, all eigenvalues of $A^\Sigma$ lie in $(-\infty, k]$; $k$ occurs with multiplicity equal to the number of balanced components.
  • Symmetry: In unsigned bipartite graphs, spectra are symmetric about $0$; for signed graphs, sign-symmetry (existence of switching to $-\Sigma$) ensures spectral symmetry, but the converse is false.
• Perturbations and spectral gaps: Changing the sign of a single edge can shrink or widen spectral gaps, unlike in the unsigned case.
• Perron–Frobenius theorem: No direct analogue unless all edge signs are positive. For general signatures, the top eigenvector may be non-dominant and eigenvalue index does not control the spectral radius.

4. Laplacians, Distance, and Product Constructions

• Laplacians: The signed Laplacian is $L^\Sigma = D(G) - A^\Sigma$, independent of the signature in its degree definition. For Hermitian adjacencies in the signed directed case, unnormalized and normalized magnetic Laplacians are:

$$L^q_U = D_s - H^q, \qquad L^q_N = I - (D_s^{-1/2}A_s D_s^{-1/2}) \odot P^q$$

Both are positive semidefinite (Ko et al., 2022). In the unsigned undirected case, this reduces to the classical Laplacian.

• Distance matrices: For signed graphs, distinct matrices $D^{\max}$ and $D^{\min}$ are constructed by tracking maximum/minimum sign of shortest paths; in certain families, notably signed complete graphs with a negative tree, adjacency and distance matrices coincide exactly, yielding striking spectral consequences (Li et al., 2022).
• Graph products: Block constructions for the signed adjacency of graph products allow explicit determination of their spectra and integrality conditions. For the product $\Sigma_1 \circledast \Sigma_2$, the adjacency matrix admits a characteristic polynomial whose roots encode the spectra of the factor graphs and their markings (Sonar et al., 2024).

5. Effective Adjacency Matrices and Deformed Laplacians

Recent developments introduce "effective adjacency matrices" associated to deformed Laplacians—including magnetic, dilation, and signal Laplacians—enabling the mapping of complex (possibly directed, signed) graphs to equivalent symmetric, unsigned graphs while retaining crucial spectral and structural properties (Resende, 2024):

• For a given deformation (specified by group elements $T_{uv}$ on each edge), and a chosen ground-state eigenvector $f$, define edge discrepancies $\xi_f(u,v)$ capturing frustration or incompatibility.
• The effective weight,

$$w_e^{(f, \beta)}(u, v) = w(u,v) \exp[-\beta\xi_f(u,v)^2],$$

adapts the original edge weight by penalizing frustration. The effective adjacency $(A_\text{eff})_{uv} = w_e^{(f, \beta)}(u, v)$ is then symmetric, real, and nonnegative, supporting the entire undirected-graph analysis toolkit.

This construction enables existing spectral, clustering, and machine learning methods to be applied to arbitrary signed and/or directed graphs, with the Hodge-Helmholtz (gradient/curl/harmonic) decomposition providing further granularity into sources of frustration or cyclicity.

6. Extremal, Structural, and Integrality Questions

Spectral extremal graph theory adapts to the signed setting with new parameters and phenomena:

• Balanced clique number $\omega_b(\Sigma)$: Maximum order of a balanced complete subgraph.
• Frustration index $\epsilon(\Sigma)$: Minimal edge set for balance.
• Spectral bounds: For signed graphs, classical theorems (Wilf, Motzkin–Straus, Nikiforov, Turán) are generalized by replacing clique number and edge counts with their balanced and frustration-adjusted analogues (Kannan et al., 2022).

The extremal spectral radius and least eigenvalue in families of signed graphs can radically differ from the unsigned case, with sign-switching playing a central role. Integrality criteria for adjacency spectra under products require the integer-root conditions on associated polynomials derived from the block structure (Sonar et al., 2024).

7. Illustrative Examples and Special Constructions

• In simple cases, e.g., $K_2$, the signed and unsigned adjacency coincide under all-positive signing; in graphs with mixed cycles or unicyclic structures, sign-invertibility and associated spectral patterns are governed by parity and matching conditions (Osborne et al., 2023).
• For signed directed graphs, the complex Hermitian formalism allows complete enumeration of edge-type cases: single-direction, bi-directional, positive/negative, each mapped to distinct complex phases and moduli, as depicted in $9$-state diagrams (Ko et al., 2022).
• Signed graph products and their adjacency matrices can be explicitly block-constructed, with the spectrum determined by the spectra and markings of input graphs along with additional terms from the coronal polynomials (Sonar et al., 2024).

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Collectively, the theory of signed and unsigned adjacency matrices spans classical and modern spectral graph theory, algorithmic challenges in combinatorics, and foundational aspects of graph neural networks and machine learning on graphs. Recent generalizations to signed directed graphs and effective adjacency constructions further extend their scope, offering new structural, spectral, and algorithmic insights.