Directed acyclic graphs (DAGs) are used for modeling causal relationships, dependencies, and flows in various systems. However, spectral analysis becomes impractical in this setting because the eigen-decomposition of the adjacency matrix yields all eigenvalues equal to zero. This inherent property of DAGs results in an inability to differentiate between frequency components of signals on such graphs. This problem can be addressed by alternating the Fourier basis or adding edges in a DAG. However, these approaches change the physics of the considered problem. To address this limitation, we propose a graph zero-padding approach. This approach involves augmenting the original DAG with additional vertices that are connected to the existing structure. The added vertices are characterized by signal values set to zero. The proposed technique enables the spectral evaluation of system outputs on DAGs (in almost all cases), that is the computation of vertex-domain convolution without the adverse effects of aliasing due to changes in a graph structure, with the ultimate goal of preserving the output of the system on a graph as if the changes in the graph structure were not done.

翻译：有向无环图（DAG）广泛用于建模各类系统中的因果关系、依赖关系和流动过程。然而，在此场景下谱分析存在实际困难，因为邻接矩阵的特征分解会产生全零特征值。DAG的这一固有特性导致无法区分此类图上信号的频率分量。现有方法可通过交替傅里叶基或在DAG中添加边来解决该问题，但这些方法会改变所研究问题的物理本质。为突破这一限制，我们提出图零填充方法。该方法通过向原始DAG添加与原结构相连的额外顶点来增强图结构，其中新增顶点的信号值设为零。该技术能够实现对DAG上系统输出的谱评估（在绝大多数情况下），即在保持图结构变化前系统输出的前提下，计算顶点域卷积且避免因图结构改变产生的混叠效应。