Geodesic
Distance spectral radius of trees with fixed maximum degree
The electronic journal of linear algebra, Tome 20 (2010), pp. 168-179.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Distance energy is a newly introduced molecular graph-based analog of the total $\pi $-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 P n has the maximal distance spectral radius among trees on n vertices. Among the trees with a fixed maximum degree $\Delta $, we prove that the broom B n,$\Delta $(consisting of a star S $\Delta +1$ and a path of length n - $\Delta - 1$ 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.
Classification : 05C05, 05C12
Keywords: distance matrix, distance spectral radius, broom graph, maximum degree 
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     author = {Stevanovic, Dragan and Ilic, Aleksandar},
     title = {Distance spectral radius of trees with fixed maximum degree},
     journal = {The electronic journal of linear algebra},
     pages = {168--179},
     publisher = {mathdoc},
     volume = {20},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2010__20__a41/}
}
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Stevanovic, Dragan; Ilic, Aleksandar. Distance spectral radius of trees with fixed maximum degree. The electronic journal of linear algebra, Tome 20 (2010), pp. 168-179. http://geodesic.mathdoc.fr/item/ELA_2010__20__a41/