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Abstract
With the growing application of undirected graphs for signal/image processing on graphs and distributed machine learning, we demonstrate that the shiftenabled condition is as necessary for undirected graphs as it is for directed graphs.It has recently been shown that, contrary to the widespread belief that a
shiftenabled condition (necessary for any shiftinvariant filter to be representable by a graph shift matrix) can be ignored because any nonshiftenabled matrix can be converted to a shiftenabled matrix, such a conversion in general may not hold for a directed graph with nonsymmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shiftenabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via examples, that a nonshiftenabled matrix cannot be converted to a shiftenabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.
shiftenabled condition (necessary for any shiftinvariant filter to be representable by a graph shift matrix) can be ignored because any nonshiftenabled matrix can be converted to a shiftenabled matrix, such a conversion in general may not hold for a directed graph with nonsymmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shiftenabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via examples, that a nonshiftenabled matrix cannot be converted to a shiftenabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.
Original language  English 

Number of pages  12 
Journal  IEEE Access 
Publication status  Accepted/In press  4 May 2021 
Keywords
 graph signal processing
 shiftenabled graphs
 shiftinvariant filter
 undirected graph
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Dive into the research topics of 'Undirected graphs: is the shiftenabled condition trivial or necessary?'. Together they form a unique fingerprint.Projects
 1 Finished

SENSIBLE: SENSors and Intelligence in BuiLt Environment (SENSIBLE) MSCA RISE
Stankovic, L., Glesk, I., Gleskova, H. & Stankovic, V.
European Commission  Horizon 2020
1/01/17 → 31/12/20
Project: Research