Characterization of Erd?s matrices by their zero entries
Bistochastic matrix; Erd?s matrix; Frobenius norm; Inner and outer (permutations and) traces; RCDS matrix; Skeleton
An Erd?s matrix E is a bistochastic matrix whose sum of squares of entries (Frobenius norm squared) equals its maxtrace (maximum value of the trace of ?E, as ? varies over permutation matrices). We characterize all Erd?s E by the patterns of their zero entries; showing that each such skeleton has at most one E. We present an algorithm to find all n Erd?s matrices, which finds them up to n?5 quickly and also size n=6. We further show some presently known RCDS matrices (E in which the trace of ?E remains constant across all the permutations that avoid every zero-entry position in E) to be Erd?s. 2026 Elsevier Inc.
Karmakar, Priyanka; Krishna, Hariram; Pal, Souvik; Teja, G. Krishna
Linear Algebra and Its Applications;Volume;739;pp.154-169
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https://doi.org/10.1016/j.laa.2026.03.010 https://www.scopus.com/pages/publications/105033551468?origin=resultslist
Karmakar P., ICTS-TIFR, Bangalore, 560 089, India; Krishna H., Department of Mathematics, IISc, Bangalore, 560 012, India; Pal S., Department of Sciences and Humanities, Christ University, Bangalore, 560 074, India; Teja G.K., Stat. Math. Unit, Indian Statistical Institute Bangalore Center, Bangalore, 560 059, India