Authors: Stevanović, Dragan 
Ilić, Aleksandar
Title: Distance spectral radius of trees with fixed maximum degree
Journal: Electronic Journal of Linear Algebra
Volume: 20
First page: 168
Last page: 179
Issue Date: 16-Aug-2010
Rank: M21
ISSN: 1081-3810
URL: https://www.math.technion.ac.il/iic/ela/ela-articles/articles/vol20_pp168-179.pdf
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 path Pn has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree Δ, we prove that the broom Bn,Δ (consisting of a star SΔ+1 and a path of length n - Δ - 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.
Keywords: Broom graph | Distance matrix | Distance spectral radius | Maximum degree
Publisher: International Linear Algebra Society
Project: Serbian Ministry of Science and Technological Development, Research Grant 144007 and144015G
Slovenian Agency for Research, program P1-0285

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