Shift-enabled condition is necessary even for symmetric shift matrices

In a 2013 paper by Sandryhaila and Moura, the authors introduced a condition (herein we will call it shift-enabled condition) that any shift invariant filter can be represented by the shift matrix if the condition is satisfied. In the same, the authors also argued that shift-enabled condition can be ignored as any non-shift-enabled matrix can be converted to a shift-enabled one. In our prior work, we proved that such conversion in general may not hold for a directed graph with non-symmetric shift matrix. This letter will focus on undirected graphs where shift matrix is generally symmetric. Though the shift matrix can be converted to satisfy shift-enabled condition, the converted matrix is not associated with the original graph, making the conversion moot. Finally, some potential methods which preserving main graph topologies to convert graph shift matrices will be introduced. Note that these methods also do not hold for all matrices and further researches on shift enabled conditions are needed.