{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,7,11]],"date-time":"2023-07-11T07:36:17Z","timestamp":1689060977541},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T00:00:00Z","timestamp":1546300800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Considered are combinatorially symmetric matrices, whose graph is a given tree, in view of the fact recent analysis shows that the geometric multiplicity theory for the eigenvalues of such matrices closely parallels that for real symmetric (and complex Hermitian) matrices. In contrast to the real symmetric case, it is shown that (a) the smallest example (13 vertices) of a tree and multiplicity list (3, 3, 3, 1, 1, 1, 1) meeting standard necessary conditions that has no real symmetric realizations does have a diagonalizable realization and for arbitrary prescribed (real and multiple) eigenvalues, and (b) that all trees with diameter &lt; 8 are geometrically di-minimal (i.e., have diagonalizable realizations with as few of distinct eigenvalues as the diameter). This re-raises natural questions about multiplicity lists that proved subtly false in the real symmetric case. What is their status in the geometric multiplicity list case?<\/jats:p>","DOI":"10.1515\/spma-2019-0025","type":"journal-article","created":{"date-parts":[[2019,12,14]],"date-time":"2019-12-14T18:08:26Z","timestamp":1576346906000},"page":"316-326","source":"Crossref","is-referenced-by-count":4,"title":["Diagonalizable matrices whose graph is a tree: the minimum number of distinct eigenvalues and the feasibility of eigenvalue assignments"],"prefix":"10.1515","volume":"7","author":[{"given":"Carlos M.","family":"Saiago","sequence":"first","affiliation":[{"name":"Centro de Matem\u00e1tica e Aplica\u00e7\u00f5es (CMA\/FCT\/UNL) , and Departamento de Matem\u00e1tica , Faculdade de Ci\u00eancias e Tecnologia da Universidade NOVA de Lisboa , 2829-516 Quinta da Torre, Portugal"}]}],"member":"374","published-online":{"date-parts":[[2019,12,13]]},"reference":[{"key":"2021102520013774642_j_spma-2019-0025_ref_001_w2aab3b7c51b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] C.R. Johnson and A. Leal-Duarte. The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree. Linear and Multilinear Algebra 46:139\u2013144 (1999).","DOI":"10.1080\/03081089908818608"},{"key":"2021102520013774642_j_spma-2019-0025_ref_002_w2aab3b7c51b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] A. Leal-Duarte and C.R. Johnson. On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree. Mathematical Inequalities and Applications 5(2):175\u2013180 (2002).","DOI":"10.7153\/mia-05-19"},{"key":"2021102520013774642_j_spma-2019-0025_ref_003_w2aab3b7c51b1b6b1ab1ab3Aa","doi-asserted-by":"crossref","unstructured":"[3] C.R. Johnson and A. Leal-Duarte. On the possible multiplicities of the eigenvalues of an Hermitian matrix whose graph is a given tree. Linear Algebra and its Applications 348:7\u201321 (2002).","DOI":"10.1016\/S0024-3795(01)00522-5"},{"key":"2021102520013774642_j_spma-2019-0025_ref_004_w2aab3b7c51b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] C.R. Johnson, A. Leal-Duarte, C.M. Saiago, B.D. Sutton, and A.J. Witt. On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a given graph. Linear Algebra and its Applications 363:147\u2013159 (2003).","DOI":"10.1016\/S0024-3795(01)00589-4"},{"key":"2021102520013774642_j_spma-2019-0025_ref_005_w2aab3b7c51b1b6b1ab1ab5Aa","doi-asserted-by":"crossref","unstructured":"[5] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars. Linear Algebra and its Applications 373:311\u2013330 (2003).","DOI":"10.1016\/S0024-3795(03)00582-2"},{"key":"2021102520013774642_j_spma-2019-0025_ref_006_w2aab3b7c51b1b6b1ab1ab6Aa","doi-asserted-by":"crossref","unstructured":"[6] C.R. Johnson and C.M. Saiago. Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree. Linear and Multilinear Algebra 56(4):357\u2013380 (2008).","DOI":"10.1080\/03081080600597668"},{"key":"2021102520013774642_j_spma-2019-0025_ref_007_w2aab3b7c51b1b6b1ab1ab7Aa","doi-asserted-by":"crossref","unstructured":"[7] C.R. Johnson, B.D. Sutton, and A. Witt. Implicit construction of multiple eigenvalues for trees. Linear and Multilinear Algebra 57(4):409\u2013420 (2009).","DOI":"10.1080\/03081080701852756"},{"key":"2021102520013774642_j_spma-2019-0025_ref_008_w2aab3b7c51b1b6b1ab1ab8Aa","doi-asserted-by":"crossref","unstructured":"[8] C.R. Johnson, J. Nuckols, and C. Spicer. The implicit construction of multiplicity lists for classes of trees and verification of some conjectures. Linear Algebra and its Applications 438(5):1990\u20132003 (2013).","DOI":"10.1016\/j.laa.2012.11.010"},{"key":"2021102520013774642_j_spma-2019-0025_ref_009_w2aab3b7c51b1b6b1ab1ab9Aa","doi-asserted-by":"crossref","unstructured":"[9] C.R. Johnson, A.A. Li, and A.J. Walker. Ordered multiplicity lists for eigenvalues of symmetric matrices whose graph is a linear tree. Discrete Mathematics 333:39\u201355 (2014).","DOI":"10.1016\/j.disc.2014.04.030"},{"key":"2021102520013774642_j_spma-2019-0025_ref_010_w2aab3b7c51b1b6b1ab1ac10Aa","doi-asserted-by":"crossref","unstructured":"[10] C.R. Johnson and C.M. Saiago. Diameter minimal trees. Linear and Multilinear Algebra 64(3):557\u2013571 (2016).","DOI":"10.1080\/03081087.2015.1057097"},{"key":"2021102520013774642_j_spma-2019-0025_ref_011_w2aab3b7c51b1b6b1ab1ac11Aa","doi-asserted-by":"crossref","unstructured":"[11] S.P. Buckley, J.G. Corliss, C.R. Johnson, C.A. Lombard\u00eda, and C.M. Saiago. Questions, conjectures, and data about multiplicity lists for trees. Linear Algebra and its Applications, 511:72\u2013109 (2016).","DOI":"10.1016\/j.laa.2016.08.002"},{"key":"2021102520013774642_j_spma-2019-0025_ref_012_w2aab3b7c51b1b6b1ab1ac12Aa","doi-asserted-by":"crossref","unstructured":"[12] C.R. Johnson and Y. Zhang. Multiplicity lists for symmetric matrices whose graphs have few missing edges. Linear Algebra and its Applications, 540:221\u2013233 (2018).","DOI":"10.1016\/j.laa.2017.11.032"},{"key":"2021102520013774642_j_spma-2019-0025_ref_013_w2aab3b7c51b1b6b1ab1ac13Aa","doi-asserted-by":"crossref","unstructured":"[13] C.R. Johnson and C.M. Saiago. Eigenvalues, Multiplicities and Graphs. Cambridge Tracts in Mathematics, Cambridge University Press, 2018.","DOI":"10.1017\/9781316155158"},{"key":"2021102520013774642_j_spma-2019-0025_ref_014_w2aab3b7c51b1b6b1ab1ac14Aa","doi-asserted-by":"crossref","unstructured":"[14] C.R. Johnson and C.M. Saiago. Geometric Parter-Wiener, etc. Theory. Linear Algebra and its Applications, 537:332\u2013347 (2018).","DOI":"10.1016\/j.laa.2017.09.035"},{"key":"2021102520013774642_j_spma-2019-0025_ref_015_w2aab3b7c51b1b6b1ab1ac15Aa","doi-asserted-by":"crossref","unstructured":"[15] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree. Linear Algebra and its Applications 559:1\u201310 (2018).","DOI":"10.1016\/j.laa.2018.08.033"},{"key":"2021102520013774642_j_spma-2019-0025_ref_016_w2aab3b7c51b1b6b1ab1ac16Aa","doi-asserted-by":"crossref","unstructured":"[16] C.R. Johnson, C. Jordan-Squire, and D.A. Sher. Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree. Discrete Applied Mathematics 158(6):681\u2013691 (2010).","DOI":"10.1016\/j.dam.2009.11.009"},{"key":"2021102520013774642_j_spma-2019-0025_ref_017_w2aab3b7c51b1b6b1ab1ac17Aa","unstructured":"[17] C.R. Johnson, J. Lettie, S. Mack-Crane, and A. Szabelska. Branch duplication in trees: Uniqueness of seed and enumeration of seeds. in manuscript"},{"key":"2021102520013774642_j_spma-2019-0025_ref_018_w2aab3b7c51b1b6b1ab1ac18Aa","doi-asserted-by":"crossref","unstructured":"[18] C.R. Johnson, A. Leal-Duarte, and C.M. Saiago. The Parter-Wiener theorem: refinement and generalization. SIAM Journal on Matrix Analysis and Applications 25(2):352\u2013361 (2003).","DOI":"10.1137\/S0895479801393320"},{"key":"2021102520013774642_j_spma-2019-0025_ref_019_w2aab3b7c51b1b6b1ab1ac19Aa","doi-asserted-by":"crossref","unstructured":"[19] F. Barioli and S.M. Fallat. On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices. Electronic Journal of Linear Algebra 11:41\u201350 (2004).","DOI":"10.13001\/1081-3810.1120"},{"key":"2021102520013774642_j_spma-2019-0025_ref_020_w2aab3b7c51b1b6b1ab1ac20Aa","unstructured":"[20] R. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, New York, 2nd Edition, 2013."},{"key":"2021102520013774642_j_spma-2019-0025_ref_021_w2aab3b7c51b1b6b1ab1ac21Aa","doi-asserted-by":"crossref","unstructured":"[21] S. Parter. On the eigenvalues and eigenvectors of a class of matrices. Journal of the Society for Industrial and Applied Mathematics 8:376\u2013388 (1960).","DOI":"10.1137\/0108024"},{"key":"2021102520013774642_j_spma-2019-0025_ref_022_w2aab3b7c51b1b6b1ab1ac22Aa","doi-asserted-by":"crossref","unstructured":"[22] G. Wiener. Spectral multiplicity and splitting results for a class of qualitative matrices. 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