Distance spectral radius of trees with fixed maximum degree

Abstract

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 ∆, we prove that the broom B n,∆ (consisting of a star S ∆+1 and 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.

A note on distance spectral radius of trees

“…Graphs with minimal and/or maximal distance spectral radius have been determined for various classes of graphs, see, e.g., [2,4,[6][7][8][9][10].…”

Graph functions maximized on a path

“…As it turns out such problem has been partially solved a while ago by Ruzieh and Powers [9], who showed that the largest eigenvalue of the distance matrix of a connected graph G of order n is maximal if G is a path. The complete solution, however, was given more recently by Stevanović and Ilić [10]. [9,10].)…”

On the second largest distance eigenvalue

“…Ruzieh and Powers [10] showed that the path P n is the unique n-vertex connected graph with maximal first distance eigenvalue, while the complete graph K n is the unique n-vertex connected graph with minimal first distance eigenvalue. Among others, Stevanović and Ilić [11] showed that the star S n is the unique n-vertex tree with minimal first distance eigenvalue. The extremal graphs with maximal or minimal first distance eigenvalues may be found in, e.g., [1,9,12,13,15].…”