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On the maximal energy tree with two maximum degree vertices

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
435
Issue
9
Identifiers
DOI: 10.1016/j.laa.2011.04.029
Keywords
  • Graph Energy
  • Tree
  • Coulson Integral Formula
Disciplines
  • Chemistry

Abstract

Abstract For a simple graph G, the energy E ( G ) is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix.For Δ ⩾ 3 and t ⩾ 3 , denote by T a ( Δ , t ) (or simply T a ) the tree formed from a path P t on t vertices by attaching Δ - 1 P 2 ’s on each end of the path P t , and T b ( Δ , t ) (or simply T b ) the tree formed from P t + 2 by attaching Δ - 1 P 2 ’s on an end of the P t + 2 and Δ - 2 P 2 ’s on the vertex next to the end.In Li et al.(2009) [16] proved that among trees of order n with two vertices of maximum degree Δ , the maximal energy tree is either the graph T a or the graph T b , where t = n + 4 - 4 Δ ⩾ 3 .However, they could not determine which one of T a and T b is the maximal energy tree.This is because the quasi-order method is invalid for comparing their energies.In this paper, we use a new method to determine the maximal energy tree.It turns out that things are more complicated.We prove that the maximal energy tree is T b for Δ ⩾ 7 and any t ⩾ 3 , while the maximal energy tree is T a for Δ = 3 and any t ⩾ 3 .Moreover, for Δ = 4 , the maximal energy tree is T a for all t ⩾ 3 but one exception that t = 4 , for which T b is the maximal energy tree.For Δ = 5 , the maximal energy tree is T b for all t ⩾ 3 but 44 exceptions that t is both odd and 3 ⩽ t ⩽ 89 , for which T a is the maximal energy tree.For Δ = 6 , the maximal energy tree is T b for all t ⩾ 3 but three exceptions that t = 3 , 5 , 7 , for which T a is the maximal energy tree.One can see that for most cases of Δ , T b is the maximal energy tree, Δ = 5 is a turning point, and Δ = 3 and 4 are exceptional cases, which means that for all chemical trees (whose maximum degrees are at most 4) with two vertices of maximum degree at least 3, T a has maximal energy, with only one exception T a ( 4 , 4 ) .

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