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ELA

DISTANCE SPECTRAL RADIUS OF TREES WITH FIXED MAXIMUM DEGREE

∗

DRAGAN STEVANOVI ´C† AND ALEKSANDAR ILI ´C‡

Abstract. Distance energy is a newly introduced molecular graph-based analog of the total π-electron energy, and it is defined as the sum of the absolute eigenvalues of the molecular distance matrix. For trees and unicyclic graphs, distance energy is equal to the doubled value of the distance spectral radius. In this paper, we introduce a general transformation that increases the distance spectral radius and provide an alternative proof that the pathPnhas the maximal distance spectral radius among trees on nvertices. Among the trees with a fixed maximum degree ∆, we prove that the broom Bn,∆ (consisting of a star S∆+1and a path of length n−∆−1 attached to an arbitrary pendent vertex of the star) is the unique tree that maximizes the distance spectral radius, and conjecture the structure of a tree which minimizes the distance spectral radius. As a first step towards this conjecture, we characterize the starlike trees with the minimum distance spectral radius.

Key words. Distance matrix, Distance spectral radius, Broom graph, Maximum degree.

AMS subject classifications.05C05, 05C12.

∗Received by the editors June 4, 2009. Accepted for publication February 17, 2010. Handling Editor: Stephen J. Kirkland.

†University of Primorska—FAMNIT, Glagoljaˇska 8, 6000 Koper, Slovenia, and Mathematical Institute, Serbian Academy of Science and Arts, Knez Mihajlova 36, 11000 Belgrade, Serbia (dra- [email protected]).

‡Faculty of Sciences and Mathematics, University of Niˇs, Viˇsegradska 33, 18000 Niˇs, Serbia (cor- responding author, [email protected]).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 20, pp. 168-179, March 2010

http://math.technion.ac.il/iic/ela

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