{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T03:19:24Z","timestamp":1762053564676,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T00:00:00Z","timestamp":1654732800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Undergraduate Innovation Training Program of Xinjiang University","award":["202110755092","2021D01C069","2019Q016","12161085"],"award-info":[{"award-number":["202110755092","2021D01C069","2019Q016","12161085"]}]},{"name":"Natural Science Foundation of Xinjiang Province","award":["202110755092","2021D01C069","2019Q016","12161085"],"award-info":[{"award-number":["202110755092","2021D01C069","2019Q016","12161085"]}]},{"name":"Youth Talent Project of Xinjiang Province","award":["202110755092","2021D01C069","2019Q016","12161085"],"award-info":[{"award-number":["202110755092","2021D01C069","2019Q016","12161085"]}]},{"name":"National Natural Science Foundation of China","award":["202110755092","2021D01C069","2019Q016","12161085"],"award-info":[{"award-number":["202110755092","2021D01C069","2019Q016","12161085"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let Pn be a pentagonal chain with 2n pentagons in which two pentagons with two edges in common can be regarded as adding one vertex and two edges to a hexagon. Thus, the linear pentagonal derivation chains QPn represent the graph obtained by attaching four-membered rings to every two pentagons of Pn. In this article, the Laplacian spectrum of QPn consisting of the eigenvalues of two symmetric matrices is determined. Next, the formulas for two graph invariants that can be represented by the Laplacian spectrum, namely, the Kirchhoff index and the number of spanning trees, are studied. Surprisingly, the Kirchhoff index is almost one half of the Wiener index of a linear pentagonal derivation chain QPn.<\/jats:p>","DOI":"10.3390\/axioms11060278","type":"journal-article","created":{"date-parts":[[2022,6,9]],"date-time":"2022-06-09T10:49:14Z","timestamp":1654771754000},"page":"278","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain"],"prefix":"10.3390","volume":"11","author":[{"given":"Yue","family":"Tu","sequence":"first","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"given":"Xiaoling","family":"Ma","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"given":"Yuqing","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]},{"given":"Junyu","family":"Ren","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Sciences, Xinjiang University, Urumqi 830017, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. 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