Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Adjacency-Operator Formulation)

Classical spectral graph theory and graph signal processing rely on a symmetry principle: undirected graphs induce symmetric (self-adjoint) adjacency/Laplacian operators, yielding orthogonal eigenbases and energy-preserving Fourier expansions.Real-world networks are typically directed and hence asymmetric, producing non-self-adjoint and frequently non-normal operators for which orthogonality fails and spectral coordinates can be ill-conditioned.In this paper we develop an original harmonic-analysis framework for directed networks centered on the adjacency operator.We propose a Biorthogonal Graph Fourier Transform (BGFT) built from left/right eigenvectors, formulate directed -frequency‖ and filtering in the non-Hermitian setting, and quantify how asymmetry and non-normality affect stability via condition numbers and a departurefrom-normality functional.We prove exact synthesis/analysis identities under diagonalizability, establish sampling-and-reconstruction guarantees for BGFT-bandlimited signals, and derive perturbation/stability bounds that explain why naive orthogonal-GFT assumptions break down on non-normal directed graphs.We provide a reproducible protocol to compare undirected versus directed cycles and a perturbed directed cycle, illustrating when orthogonal-GFT assumptions fail and how BGFT behaves across asymmetric regimes.