{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:39:51Z","timestamp":1760060391644,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,8,22]],"date-time":"2025-08-22T00:00:00Z","timestamp":1755820800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11961040"],"award-info":[{"award-number":["11961040"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let G be a connected graph. The Steiner k-Wiener index SWk(G) of graph G is defined as SWk(G)=\u2211S\u2286V(G),|S|=kdG(S), where dG(S) represents the minimum size of a connected subgraph of G that connects S. Using some graph operations, we obtain the minimum value and the second minimum value of the Steiner k-Wiener index for cacti with order n and t cycles, and we characterize the corresponding extremal graphs by exploiting structural symmetries.<\/jats:p>","DOI":"10.3390\/sym17091371","type":"journal-article","created":{"date-parts":[[2025,8,22]],"date-time":"2025-08-22T07:41:45Z","timestamp":1755848505000},"page":"1371","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Steiner k-Wiener Index of Cacti"],"prefix":"10.3390","volume":"17","author":[{"given":"Chengye","family":"Xu","sequence":"first","affiliation":[{"name":"School of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China"}]},{"given":"Mengmeng","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. 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Lett."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1371\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:33:24Z","timestamp":1760034804000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/9\/1371"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,22]]},"references-count":18,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["sym17091371"],"URL":"https:\/\/doi.org\/10.3390\/sym17091371","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,8,22]]}}}