Spectral GNNs, like ChebyNet, are limited by heterophily and over-smoothing due to their static, low-pass filter design. This work investigates the "Adaptive Orthogonal Polynomial Filter" (AOPF) class as a solution. We introduce two models operating in the [-1, 1] domain: 1) `L-JacobiNet`, the adaptive generalization of `ChebyNet` with learnable alpha, beta shape parameters, and 2) `S-JacobiNet`, a novel baseline representing a LayerNorm-stabilized static `ChebyNet`. Our analysis, comparing these models against AOPFs in the [0, infty) domain (e.g., `LaguerreNet`), reveals critical, previously unknown trade-offs. We find that the [0, infty) domain is superior for modeling heterophily, while the [-1, 1] domain (Jacobi) provides superior numerical stability at high K (K>20). Most significantly, we discover that `ChebyNet`'s main flaw is stabilization, not its static nature. Our static `S-JacobiNet` (ChebyNet+LayerNorm) outperforms the adaptive `L-JacobiNet` on 4 out of 5 benchmark datasets, identifying `S-JacobiNet` as a powerful, overlooked baseline and suggesting that adaptation in the [-1, 1] domain can lead to overfitting.

Spectral Graph Neural Networks (GNNs) based on polynomial filters, such as ChebyNet, suffer from two critical limitations: 1) performance collapse on "heterophilic" graphs and 2) performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters. In this work, we propose `KrawtchoukNet`, a GNN filter based on the discrete Krawtchouk polynomials. We demonstrate that `KrawtchoukNet` provides a unified solution to both problems through two key design choices. First, by fixing the polynomial's domain N to a small constant (e.g., N=20), we create the first GNN filter whose recurrence coefficients are \textit{inherently bounded}, making it exceptionally robust to over-smoothing (achieving SOTA results at K=10). Second, by making the filter's shape parameter p learnable, the filter adapts its spectral response to the graph data. We show this adaptive nature allows `KrawtchoukNet` to achieve SOTA performance on challenging heterophilic benchmarks (Texas, Cornell), decisively outperforming standard GNNs like GAT and APPNP.

Graph Neural Networks (GNNs) based on spectral filters, such as the Adaptive Orthogonal Polynomial Filter (AOPF) class (e.g., LaguerreNet), have shown promise in unifying the solutions for heterophily and over-smoothing. However, these single-filter models suffer from a "compromise" problem, as their single adaptive parameter (e.g., alpha) must learn a suboptimal, averaged response across the entire graph spectrum. In this paper, we propose DualLaguerreNet, a novel GNN architecture that solves this by introducing "Decoupled Spectral Flexibility." DualLaguerreNet splits the graph Laplacian into two operators, L_low (low-frequency) and L_high (high-frequency), and learns two independent, adaptive Laguerre polynomial filters, parameterized by alpha_1 and alpha_2, respectively. This work, however, uncovers a deeper finding. While our experiments show DualLaguerreNet's flexibility allows it to achieve state-of-the-art results on complex heterophilic tasks (outperforming LaguerreNet), it simultaneously underperforms on simpler, homophilic tasks. We identify this as a fundamental "Flexibility-Stability Trade-off". The increased parameterization (2x filter parameters and 2x model parameters) leads to overfitting on simple tasks, demonstrating that the "compromise" of simpler models acts as a crucial regularizer. This paper presents a new SOTA architecture for heterophily while providing a critical analysis of the bias-variance trade-off inherent in adaptive GNN filter design.

Spectral Graph Neural Networks (GNNs) have achieved state-of-the-art results by defining graph convolutions in the spectral domain. A common approach, popularized by ChebyNet, is to use polynomial filters based on continuous orthogonal polynomials (e.g., Chebyshev). This creates a theoretical disconnect, as these continuous-domain filters are applied to inherently discrete graph structures. We hypothesize this mismatch can lead to suboptimal performance and fragility to hyperparameter settings. In this paper, we introduce MeixnerNet, a novel spectral GNN architecture that employs discrete orthogonal polynomials -- specifically, the Meixner polynomials $M_k(x; β, c)$. Our model makes the two key shape parameters of the polynomial, beta and c, learnable, allowing the filter to adapt its polynomial basis to the specific spectral properties of a given graph. We overcome the significant numerical instability of these polynomials by introducing a novel stabilization technique that combines Laplacian scaling with per-basis LayerNorm. We demonstrate experimentally that MeixnerNet achieves competitive-to-superior performance against the strong ChebyNet baseline at the optimal K = 2 setting (winning on 2 out of 3 benchmarks). More critically, we show that MeixnerNet is exceptionally robust to variations in the polynomial degree K, a hyperparameter to which ChebyNet proves to be highly fragile, collapsing in performance where MeixnerNet remains stable.