{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:46:53Z","timestamp":1760060813218,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T00:00:00Z","timestamp":1758499200000},"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":["12261071","2025-ZJ-902T"],"award-info":[{"award-number":["12261071","2025-ZJ-902T"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Qinghai Province","award":["12261071","2025-ZJ-902T"],"award-info":[{"award-number":["12261071","2025-ZJ-902T"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let \u03c7n\u22121(\u03c3) denote the irreducible character of the symmetric group Sn corresponding to the partition (n\u22121,1). For an n\u00d7n matrix M=(mi,j), we denote its (n\u22121)-th immanant by dn\u22121(M). Let G be a simple connected graph and let L(G) and Q(G) denote the Laplacian matrix and the signless Laplacian matrix of G, respectively. The (n\u22121)-th Laplacian (respectively, signless Laplacian) immanantal polynomial of G is defined as dn\u22121(xI\u2212L(G)) (respectively, dn\u22121(xI\u2212Q(G))). In this paper, we partially resolve Chan\u2019s open problem by establishing that the broom graph minimizes dn\u22121(L(T)) among all trees with given diameter. Furthermore, we give combinatorial expressions for the first five coefficients of the (n\u22121)-th Laplacian immanantal polynomial dn\u22121(xI\u2212L(G)). We also investigate the characterizing properties of this polynomial and present several graphs that are uniquely determined by it. Additionally, for the (n\u22121)-th signless Laplacian immanantal polynomial dn\u22121(xI\u2212Q(G)), we show that the multiplicity of root 1 is bounded below by the star degree of G.<\/jats:p>","DOI":"10.3390\/axioms14090716","type":"journal-article","created":{"date-parts":[[2025,9,22]],"date-time":"2025-09-22T17:08:17Z","timestamp":1758560897000},"page":"716","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The (n-1)-th Laplacian Immanantal Polynomials of Graphs"],"prefix":"10.3390","volume":"14","author":[{"given":"Wenwei","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2488-9775","authenticated-orcid":false,"given":"Tingzeng","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"},{"name":"Qinghai Institute of Applied Mathematics, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xianyue","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"},{"name":"School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Valiant, L.G. (1979). The complexity of computing the permanent. Theoret. Comput. Sci., 189\u2013201.","DOI":"10.1016\/0304-3975(79)90044-6"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1023","DOI":"10.1137\/S0097539798367880","article-title":"The computational complexity of immanants","volume":"30","year":"2000","journal-title":"SIAM J. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1137\/0405036","article-title":"Laplacian permanents of trees","volume":"5","author":"Botti","year":"1992","journal-title":"SIAM J. Discret. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"520","DOI":"10.2307\/1969645","article-title":"Determinant theory in finite factors","volume":"55","author":"Fuglede","year":"1952","journal-title":"Ann. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/0024-3795(87)90159-5","article-title":"A Fischer inequality for the second immanant","volume":"87","author":"Grone","year":"1987","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1080\/0025570X.1969.11975950","article-title":"Determinants, permanents and bipartite graphs","volume":"42","author":"Harary","year":"1969","journal-title":"Math. Mag."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/andp.18471481202","article-title":"Ueber die Aufl\u00f6sung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Str\u00f6me gef\u00fchrt wird","volume":"148","author":"Kirchhoff","year":"1847","journal-title":"Ann. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.laa.2003.11.033","article-title":"Immanantal invariants of graphs","volume":"401","author":"Merris","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1137\/S0895479897318423","article-title":"Immanant inequalities for Laplacians of trees","volume":"21","author":"Chan","year":"1999","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0012-365X(84)90127-4","article-title":"Permanent of the Laplacian matrix of trees and bipartite graphs","volume":"48","author":"Brualdi","year":"1984","journal-title":"Discret. Math."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Brouwer, A.E., and Haemers, W.H. (2011). Spectra of Graphs, Springer.","DOI":"10.1007\/978-1-4614-1939-6"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D.M., Rowlinson, P., and Simi\u0107, S. (2010). An Introduction to the Theory of Graph Spectra, Cambridge University Press.","DOI":"10.1017\/CBO9780511801518"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Shi, Y., Dehmer, M., Li, X., and Gutman, I. (2017). On the permanental polynomials of graphs. Graph Polynomials, Chapman & Hall\/CRC.","DOI":"10.1201\/9781315367996"},{"key":"ref_14","first-page":"38","article-title":"The coefficients of the immanantal polynomial","volume":"339","author":"Yu","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0024-3795(85)90281-2","article-title":"Permanental roots and star degree of a graph","volume":"64","author":"Faria","year":"1985","journal-title":"Linear Algebra Appl."},{"key":"ref_16","first-page":"105624","article-title":"Signless Laplacian polynomial and characteristic polynomial of a graph","volume":"1","author":"Ramane","year":"2013","journal-title":"Discret. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"114105","DOI":"10.1016\/j.disc.2024.114105","article-title":"On the second immanantal polynomials of graphs","volume":"347","author":"Wu","year":"2024","journal-title":"Discret. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s00373-023-02710-3","article-title":"On the roots of (signless) laplacian permanental polynomials of graphs","volume":"39","author":"Wu","year":"2023","journal-title":"Graphs Combin."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"14","DOI":"10.47443\/dml.2022.005","article-title":"On the permanental polynomial and permanental sum of signed graphs","volume":"10","author":"Tang","year":"2022","journal-title":"Discret. Math. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1137\/0607056","article-title":"The second immanantal polynomial and the centroid of a graph","volume":"7","author":"Merris","year":"1986","journal-title":"SIAM J. Algebr. Discret. Methods"},{"key":"ref_21","unstructured":"Sagan, B. (1991). The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Wadsworth & Books\/Cole."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/0024-3795(86)90124-2","article-title":"A bound for the permanent of the Laplacian matrix","volume":"74","author":"Bapat","year":"1986","journal-title":"Linear Algebra Appl."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Brualdi, R.A., and Ryser, H.J. (1991). Combinatorial Matrix Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781107325708"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/0024-3795(88)90320-5","article-title":"Inequalities for single-hook immanants","volume":"102","author":"Johnson","year":"1988","journal-title":"Linear Algebra Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"787","DOI":"10.1007\/s00373-019-02033-2","article-title":"On the (signless) Laplacian permanental polynomials of graphs","volume":"35","author":"Liu","year":"2019","journal-title":"Graphs Comb."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1080\/03081087.2013.869592","article-title":"Per-spectral characterizations of some edge-deleted subgraphs of a complete graph","volume":"63","author":"Zhang","year":"2015","journal-title":"Linear Multilinear Algebra"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/716\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:47:13Z","timestamp":1760035633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/9\/716"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,22]]},"references-count":26,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2025,9]]}},"alternative-id":["axioms14090716"],"URL":"https:\/\/doi.org\/10.3390\/axioms14090716","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,9,22]]}}}