{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:19:31Z","timestamp":1760235571680,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2021,9,10]],"date-time":"2021-09-10T00:00:00Z","timestamp":1631232000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100009074","name":"Huaiyin Institute of Technology","doi-asserted-by":"publisher","award":["16HGZ007"],"award-info":[{"award-number":["16HGZ007"]}],"id":[{"id":"10.13039\/501100009074","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An n\u00d7n matrix A is called eventually exponentially positive (EEP) if etA=\u2211k=0\u221etkAkk!&gt;0 for all t\u2265t0, where t0\u22650. A matrix whose entries belong to the set {+,\u2212,0} is called a sign pattern. An n\u00d7n sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n\u22643 are identified. For the n\u00d7n tridiagonal sign patterns Tn, we show that there exists exactly one MPEEP tridiagonal sign pattern Tno. Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of Tno. We also classify all PEEP star sign patterns Sn and double star sign patterns DS(n,m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.<\/jats:p>","DOI":"10.3390\/sym13091669","type":"journal-article","created":{"date-parts":[[2021,9,12]],"date-time":"2021-09-12T21:48:01Z","timestamp":1631483281000},"page":"1669","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Eventual Exponential Positivity of Some Tree Sign Patterns"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6009-2865","authenticated-orcid":false,"given":"Ber-Lin","family":"Yu","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai\u2019an 223003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongshan","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30302-4110, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sanzhang","family":"Xu","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai\u2019an 223003, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,10]]},"reference":[{"key":"ref_1","unstructured":"Horn, R.A., and Johnson, C.R. 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Lett."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1669\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:00:03Z","timestamp":1760166003000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/9\/1669"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,10]]},"references-count":10,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["sym13091669"],"URL":"https:\/\/doi.org\/10.3390\/sym13091669","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,9,10]]}}}