{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:21:50Z","timestamp":1760145710346,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"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":["12361071","11901498"],"award-info":[{"award-number":["12361071","11901498"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A signed graph \u03a3=(G,\u03c3) consists of an underlying graph G=(V,E) with a sign function \u03c3:E\u2192{\u22121,1}. Let A(\u03a3) be the adjacency matrix of \u03a3 and \u03bb1(\u03a3) denote the largest eigenvalue (index) of \u03a3. Define (Kn,H\u2212) as a signed complete graph whose negative edges induce a subgraph H. In this paper, we focus on the following question: which spanning tree T with a given number of pendant vertices makes the \u03bb1(A(\u03a3)) of the unbalanced (Kn,T\u2212) as large as possible? To answer the question, we characterize the extremal signed graph with maximum \u03bb1(A(\u03a3)) among graphs of type (Kn,T\u2212).<\/jats:p>","DOI":"10.3390\/axioms13080565","type":"journal-article","created":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T06:07:35Z","timestamp":1724134055000},"page":"565","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Maximizing the Index of Signed Complete Graphs Containing a Spanning Tree with k Pendant Vertices"],"prefix":"10.3390","volume":"13","author":[{"given":"Dan","family":"Li","sequence":"first","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Minghui","family":"Yan","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhaolin","family":"Teng","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1307\/mmj\/1028989917","article-title":"On the notion of balance in a signed graph","volume":"2","author":"Harary","year":"1953","journal-title":"Michigan Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1037\/h0046049","article-title":"Structural balance: A generalization of Heider\u2019s theory","volume":"63","author":"Cartwright","year":"1956","journal-title":"Psychol Rev."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0166-218X(82)90033-6","article-title":"Signed graphs","volume":"4","author":"Zaslavsky","year":"1982","journal-title":"Discret. Appl. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1137\/0603033","article-title":"A combinatorial proof of the all minors matrix tree theorem","volume":"3","author":"Chaiken","year":"1982","journal-title":"SIAM J. Algebr. Discret. Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/0095-8956(89)90063-4","article-title":"Biased graphs, I: Bias, balance, and gains","volume":"47","author":"Zaslavsky","year":"1989","journal-title":"J. Combin. Theory Ser. B"},{"key":"ref_6","unstructured":"Zaslavsky, T. (2013). Matrices in the theory of signed simple graphs. arXiv."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1016\/j.laa.2004.08.025","article-title":"The spectral radius of trees on k pendant vertices","volume":"395","author":"Wu","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1016\/j.laa.2005.05.022","article-title":"The spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices","volume":"408","author":"Guo","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_9","first-page":"77","article-title":"The spectral radius of tricyclic graphs with n vertices and k pendant edges","volume":"86","year":"2008","journal-title":"Ars Combin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1080\/03081080902722816","article-title":"On the spectral radius of cacti with k-pendant vertices","volume":"58","author":"Wu","year":"2010","journal-title":"Linear Multilinear Algebra"},{"key":"ref_11","first-page":"19","article-title":"Minimizing the least eigenvalue of bicyclic graphs with k pendant vertices","volume":"103","author":"Liu","year":"2012","journal-title":"Ars Combin."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.laa.2009.03.016","article-title":"The least eigenvalue of unicyclic graphs with n vertices and k pendant vertices","volume":"431","author":"Liu","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2256","DOI":"10.1016\/j.laa.2012.09.029","article-title":"On the least eigenvalue of cacti with pendant vertices","volume":"438","author":"Xing","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Brouwer, A.E., and Haemers, W.H. (2012). Spectra of Graphs, Spring.","DOI":"10.1007\/978-1-4614-1939-6"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1007\/s40314-022-01814-5","article-title":"Unbalanced signed graphs with extremal spectral radius or index","volume":"41","author":"Brunetti","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2050016","DOI":"10.1142\/S1793830920500160","article-title":"Maximizing the largest eigenvalues of signed unicyclic graphs","volume":"12","author":"Souri","year":"2020","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"128478","DOI":"10.1016\/j.amc.2023.128478","article-title":"Unbalanced signed bicyclic graphs minimizing the least eigenvalue","volume":"466","author":"Teng","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"14","DOI":"10.13001\/ela.2024.7947","article-title":"Minimizing the least eigenvalue of unbalanced signed unicyclic graphs with given girth or pendant vertices","volume":"40","author":"Yan","year":"2024","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"128814","DOI":"10.1016\/j.amc.2024.128814","article-title":"Tur\u00e1n problem for K4\u2212-free signed graphs","volume":"477","author":"Chen","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.laa.2023.10.024","article-title":"Extremal results for C3\u2212-free signed graphs","volume":"681","author":"Wang","year":"2024","journal-title":"Linear Algebra Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.laa.2024.03.019","article-title":"Spectral Tur\u00e1n problem for K5\u2212-free signed graphs","volume":"691","author":"Wang","year":"2024","journal-title":"Linear Multilinear Algebra"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2187","DOI":"10.1080\/03081087.2016.1265480","article-title":"Connected signed graphs of fixed order, size, and number of negative edges with maximal index","volume":"65","author":"Koledin","year":"2017","journal-title":"Linear Multilinear Algebra"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"393","DOI":"10.7151\/dmgt.2276","article-title":"Signed complete graphs with maximum index","volume":"40","author":"Akbari","year":"2020","journal-title":"Discuss. Math. Graph Theory"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112463","DOI":"10.1016\/j.disc.2021.112463","article-title":"Signed graphs with maximal index","volume":"344","author":"Ghorbani","year":"2021","journal-title":"Discret. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"113250","DOI":"10.1016\/j.disc.2022.113250","article-title":"Extremal spectral results related to spanning trees of signed complete graphs","volume":"346","author":"Li","year":"2023","journal-title":"Discret. Math."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2085","DOI":"10.1007\/s40995-021-01199-w","article-title":"On the signed complete graphs with maximum index","volume":"45","author":"Kafai","year":"2021","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"113860","DOI":"10.1016\/j.disc.2023.113860","article-title":"Complete signed graphs with largest maximum or smallest minimum eigenvalue","volume":"347","author":"Ghorbani","year":"2024","journal-title":"Discret. 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