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Completely positive matrices of order 5 with a nearly \widehat{CP}-graph


Abstract


A particular class of graphs, called "nearly \widehat{CP}-graphs", extending the class of \widehat{CP}-graphs is introduced. From Barioli's characterization of completely positive matrices with a book-graph, two equivalent characterizations of completely positive matrices with a nearly \widehat{CP}-graph are deduced. By means of these results, new characterizations of completely positive matrices of order 5 with a book-graph and some alternative demonstrations of known results are derived. A new characterization of completely positive matrices of order 5 with a \widehat{CP}-graph and a new elementary proof of the main result obtained by Cedolin-Salce, regarding this particular class of matrices, are shown. It is also attempted to clarify some results obtained by Xu on completely positive matrices of order 5, and it is shown by a counterexample that one of them is incorrect.

DOI Code: 10.1285/i15900932v39n1p89

Keywords: Completely positive matrices; doubly non-negative matrices; book graphs

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