  Graph-level anomaly detection has gained significant attention as it finds
applications in various domains, such as cancer diagnosis and enzyme
prediction. However, existing methods fail to capture the spectral properties
of graph anomalies, resulting in unexplainable framework design and
unsatisfying performance. In this paper, we re-investigate the spectral
differences between anomalous and normal graphs. Our main observation shows a
significant disparity in the accumulated spectral energy between these two
classes. Moreover, we prove that the accumulated spectral energy of the graph
signal can be represented by its Rayleigh Quotient, indicating that the
Rayleigh Quotient is a driving factor behind the anomalous properties of
graphs. Motivated by this, we propose Rayleigh Quotient Graph Neural Network
(RQGNN), the first spectral GNN that explores the inherent spectral features of
anomalous graphs for graph-level anomaly detection. Specifically, we introduce
a novel framework with two components: the Rayleigh Quotient learning component
(RQL) and Chebyshev Wavelet GNN with RQ-pooling (CWGNN-RQ). RQL explicitly
captures the Rayleigh Quotient of graphs and CWGNN-RQ implicitly explores the
spectral space of graphs. Extensive experiments on 10 real-world datasets show
that RQGNN outperforms the best rival by 6.74% in Macro-F1 score and 1.44% in
AUC, demonstrating the effectiveness of our framework. Our code is available at
https://github.com/xydong127/RQGNN.
