{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T21:56:24Z","timestamp":1770414984067,"version":"3.49.0"},"reference-count":66,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,5]],"date-time":"2015-06-05T00:00:00Z","timestamp":1433462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"austrian science foundation","award":["P26142"],"award-info":[{"award-number":["P26142"]}]},{"DOI":"10.13039\/501100002347","name":"BMBF","doi-asserted-by":"publisher","award":["13N12304"],"award-info":[{"award-number":["13N12304"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["61174094"],"award-info":[{"award-number":["61174094"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006606","name":"Natural Science Foundation of Tianjin","doi-asserted-by":"publisher","award":["14JCYBJC18700"],"award-info":[{"award-number":["14JCYBJC18700"]}],"id":[{"id":"10.13039\/501100006606","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Shannon entropies for networks have been widely introduced. However, entropies for weighted graphs have been little investigated. Inspired by the work due to Eagle et al., we introduce the concept of graph entropy for special weighted graphs. Furthermore, we prove extremal properties by using elementary methods of classes of weighted graphs, and in particular, the one due to Bollob\u00e1s and Erd\u00f6s, which is also called the Randi\u00b4c weight. As a result, we derived statements on dendrimers that have been proven useful for applications. Finally, some open problems are presented.<\/jats:p>","DOI":"10.3390\/e17063710","type":"journal-article","created":{"date-parts":[[2015,6,5]],"date-time":"2015-06-05T10:23:07Z","timestamp":1433499787000},"page":"3710-3723","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":32,"title":["Entropy of Weighted Graphs with Randi\u00b4c Weights"],"prefix":"10.3390","volume":"17","author":[{"given":"Zengqiang","family":"Chen","sequence":"first","affiliation":[{"name":"College of Computer and Control Engineering, Nankai University, Tianjin 300071, China"}]},{"given":"Matthias","family":"Dehmer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Universit\u00e4t der Bundeswehr M\u00fcnchen, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany"},{"name":"Department of Mechatronics and Biomedical Computer Science, UMIT, A-6060 Hall in Tyrol, Austria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0745-5641","authenticated-orcid":false,"given":"Frank","family":"Emmert-Streib","sequence":"additional","affiliation":[{"name":"Computational Medicine and Statistical Learning Laboratory, Department of Signal Processing, Tampere University of Technology, FI-33720 Tampere, Finland"},{"name":"Institute of Biosciences and Medical Technology, 33520 Tampere, Finland"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[{"name":"Center for Combinatorics and LPMC-TJKLC, Nankai University, Tianjin 300071, China"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,5]]},"reference":[{"key":"ref_1","unstructured":"Shannon, C., and Weaver, W. 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