Graph Neural Networks (GNNs) have received extensive research attention for their promising performance in graph machine learning. Despite their extraordinary predictive accuracy, existing approaches, such as GCN and GPRGNN, are not robust in the face of homophily changes on test graphs, rendering these models vulnerable to graph structural attacks and with limited capacity in generalizing to graphs of varied homophily levels. Although many methods have been proposed to improve the robustness of GNN models, the majority of these techniques are restricted to the spatial domain and employ complicated defense mechanisms, such as learning new graph structures or calculating edge attention. In this paper, we study the problem of designing simple and robust GNN models in the spectral domain. We propose EvenNet, a spectral GNN corresponding to an even-polynomial graph filter. Based on our theoretical analysis in both spatial and spectral domains, we demonstrate that EvenNet outperforms full-order models in generalizing across homophilic and heterophilic graphs, implying that ignoring odd-hop neighbors improves the robustness of GNNs. We conduct experiments on both synthetic and real-world datasets to demonstrate the effectiveness of EvenNet. Notably, EvenNet outperforms existing defense models against structural attacks without introducing additional computational costs and maintains competitiveness in traditional node classification tasks on homophilic and heterophilic graphs.

Polynomial-based learnable spectral graph neural networks (GNNs) utilize polynomial to approximate graph convolutions and have achieved impressive performance on graphs. Nevertheless, there are three progressive problems to be solved. Some models use polynomials with better approximation for approximating filters, yet perform worse on real-world graphs. Carefully crafted graph learning methods, sophisticated polynomial approximations, and refined coefficient constraints leaded to overfitting, which diminishes the generalization of the models. How to design a model that retains the ability of polynomial-based spectral GNNs to approximate filters while it possesses higher generalization and performance?

Spectral Graph Neural Networks (SGNNs) have achieved remarkable performance in tasks such as node classification due to their ability to learn flexible filters. Typically, these filters are learned under the supervision of downstream tasks, enabling SGNNs to adapt to diverse structural patterns. However, in scenarios with limited labeled data, SGNNs often struggle to capture the optimal filter shapes, resulting in degraded performance, especially on graphs with heterophily. Meanwhile, the rapid progress of Large Language Models (LLMs) has opened new possibilities for enhancing graph learning without modifying graph structure or requiring task-specific training. In this work, we propose a novel framework that leverages LLMs to estimate the homophily level of a graph and uses this global structural prior to guide the construction of spectral filters. Specifically, we design a lightweight and plug-and-play pipeline where a small set of labeled node pairs is formatted as natural language prompts for the LLM, which then predicts the graph's homophily ratio. This estimated value informs the spectral filter basis, enabling SGNNs to adapt more effectively to both homophilic and heterophilic structures. Extensive experiments on multiple benchmark datasets demonstrate that our LLM-assisted spectral framework consistently improves performance over strong SGNN baselines.

Polynomial-based learnable spectral graph neural networks (GNNs) utilize polynomial to approximate graph convolutions and have achieved impressive performance on graphs. Nevertheless, there are three progressive problems to be solved. Some models use polynomials with better approximation for approximating filters, yet perform worse on real-world graphs. Carefully crafted graph learning methods, sophisticated polynomial approximations, and refined coefficient constraints leaded to overfitting, which diminishes the generalization of the models. How to design a model that retains the ability of polynomial-based spectral GNNs to approximate filters while it possesses higher generalization and performance?

Graph Neural Networks (GNNs) play a pivotal role in graph-based tasks for their proficiency in representation learning. Among the various GNN methods, spectral GNNs employing polynomial filters have shown promising performance on tasks involving both homophilous and heterophilous graph structures. However, The scalability of spectral GNNs on large graphs is limited because they learn the polynomial coefficients through multiple forward propagation executions during forward propagation. Existing works have attempted to scale up spectral GNNs by eliminating the linear layers on the input node features, a change that can disrupt end-to-end training, potentially impact performance, and become impractical with high-dimensional input features. To address the above challenges, we propose "Spectral Graph Neural Networks with Laplacian Sparsification (SGNN-LS)", a novel graph spectral sparsification method to approximate the propagation patterns of spectral GNNs. We prove that our proposed method generates Laplacian sparsifiers that can approximate both fixed and learnable polynomial filters with theoretical guarantees. Our method allows the application of linear layers on the input node features, enabling end-to-end training as well as the handling of raw text features. We conduct an extensive experimental analysis on datasets spanning various graph scales and properties to demonstrate the superior efficiency and effectiveness of our method. The results show that our method yields superior results in comparison with the corresponding approximated base models, especially on dataset Ogbn-papers100M(111M nodes, 1.6B edges) and MAG-scholar-C (2.8M features).

Approximation-based spectral graph neural networks, which construct graph filters with function approximation, have shown substantial performance in graph learning tasks. Despite their great success, existing works primarily employ polynomial approximation to construct the filters, whereas another superior option, namely ration approximation, remains underexplored. Although a handful of prior works have attempted to deploy the rational approximation, their implementations often involve intensive computational demands or still resort to polynomial approximations, hindering full potential of the rational graph filters. To address the issues, this paper introduces ERGNN, a novel spectral GNN with explicitly-optimized rational filter. ERGNN adopts a unique two-step framework that sequentially applies the numerator filter and the denominator filter to the input signals, thus streamlining the model paradigm while enabling explicit optimization of both numerator and denominator of the rational filter. Extensive experiments validate the superiority of ERGNN over state-of-the-art methods, establishing it as a practical solution for deploying rational-based GNNs.

In real-world applications, spectral Graph Neural Networks (GNNs) are powerful tools for processing diverse types of graphs. However, a single GNN often struggles to handle different graph types-such as homogeneous and heterogeneous graphs-simultaneously. This challenge has led to the manual design of GNNs tailored to specific graph types, but these approaches are limited by the high cost of labor and the constraints of expert knowledge, which cannot keep up with the rapid growth of graph data. To overcome these challenges, we propose AutoSGNN, an automated framework for discovering propagation mechanisms in spectral GNNs. AutoSGNN unifies the search space for spectral GNNs by integrating large language models with evolutionary strategies to automatically generate architectures that adapt to various graph types. Extensive experiments on nine widely-used datasets, encompassing both homophilic and heterophilic graphs, demonstrate that AutoSGNN outperforms state-of-the-art spectral GNNs and graph neural architecture search methods in both performance and efficiency.