Two of the most widely used methods for analysing graph data, Adjacency Spectral Embedding and Laplacian Spectral Embedding, often produce different results when applied to the same graph. Yet the structural reasons behind this disagreement remain incompletely understood. This paper provides an end-to-end account of ASE-LSE latent subspace disagreement. We first prove that the two methods produce identical latent subspaces for every embedding dimension whenever the Laplacian is a scalar multiple of the adjacency matrix, and show that this scalar relationship holds if and only if the graph is either regular or bipartite biregular. This anchor result identifies a sufficient condition for perfect agreement that pins down the floor of the disagreement spectrum and supplies the baseline for the perturbation analysis. We then prove that no maximal-disagreement graph or family of graphs exists: the disagreement is always strictly below its theoretical ceiling, and we exhibit a witness family demonstrating that no finite maximum is attainable, so the disagreement landscape has no maximiser. With both endpoints established, we derive a Regularity Departure Bound whose two terms isolate degree heterogeneity and eigengap as the primary structural factors influencing disagreement in the middle regime. Empirical validation across thousands of simulated graphs confirms the mechanisms predicted by the bound: heterogeneity pushes disagreement up, eigengap suppresses it, and their joint ratio emerges as a unified predictor of ASE-LSE disagreement, suggesting when the two embeddings can be treated as interchangeable and when they cannot.

翻译：邻接谱嵌入与拉普拉斯谱嵌入是分析图数据最广泛使用的两种方法，但应用于同一图时往往产生不同结果。然而，这种不一致背后的结构性原因尚未被完全理解。本文对ASE-LSE潜在子空间不一致性进行了端到端的阐述。我们首先证明，当且仅当拉普拉斯矩阵是邻接矩阵的标量倍数时，两种方法会对每个嵌入维度产生相同的潜在子空间，并证明这种标量关系成立当且仅当图为正则图或二部双正则图。这一锚定结果识别出完美一致性的充分条件，确定了不一致性谱的下界，并为扰动分析提供了基准。接着，我们证明不存在最大不一致性的图或图族：不一致性严格低于其理论上限，并通过一个示例图族表明无法达到有限最大值，因此不一致性景观不存在极大值点。在确立两个端点后，我们推导出正则性偏离界，其两项分别将度异质性和特征间隙隔离为影响中间区域不一致性的主要结构因素。对数千个模拟图的实证验证确认了该界所预测的机制：度异质性推高不一致性，特征间隙抑制不一致性，两者的联合比率成为ASE-LSE不一致性的统一预测因子，从而揭示两种嵌入何时可互换、何时不可互换。