  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.
