Distinguishing Resistance-equivalent Isomorphic Graphs via Spectral-based Graph Representations

Cheng-Yu Ma, Yi-Ting Hsieh, Chung-Shou Liao

Keywords: Graph Representation, Graph Isomorphism, Spectral Methods, Laplacian eigenvectors, Resistance Spectrum, Graph Neural Network

Abstract:

Graph Neural Networks (GNNs) usually face an expressive power bottleneck, limiting their capability to distinguish pairs of graphs that are structurally similar yet non-isomorphic.Very recently, Graphormer-GD, one of state-of-the-art algorithms, leveraged resistance distance to represent graph topological features and effectively solved the problem. However, the resistance distance feature fails to distinguish resistance-equivalent graphs. To address the fundamental challenge, a new representation method, using spectral features—the eigenvectors of the graph Laplacian—to capture the topological information that resistance distance overlooks, is proposed in this study, showing the discriminability between representative resistance-equivalent graphs. To demonstrate the capability of our embedding method on real world datasets, a model, called Spectral-Graphormer, integrating spectral features into the Graphormer architecture, is also proposed. The model outperforms the original Graphormer and performs comparably to Graphormer-GD, and surpasses both on the large-scale benchmark, achieving test accuracy comparable to MPNN baselines such as PNA. In particular,
rather than constructing a memory-intensive $N \times N$ resistance-distance matrix, the model builds an $N \times k$ spectral-embedding matrix, which significantly reduces memory usage—especially in large-scale graphs.
Apparently, spectral features provide a more robust and scalable structural encoding for graph representations.
The proposed approach is theoretically grounded and practically effective for distinguishing non-isomorphic graphs with similar structures, and it further demonstrates significant merits in large-scale graph tasks.

Primary Area: learning on graphs and other geometries & topologies

Submission Number: 20049