2011/03/20 by Jing Li, Xueliang Li, Li, Jing +3
Mathematics · #05C50 #05C90 #15A18 #92E10 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C50 #msc:05C90 #msc:15A18 #msc:92E10
paper · pdf · doi:10.48550/arxiv.1103.3842
16 pages
arxiv created 2011/03/29 · arxiv updated 2011/03/30
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 Ta(Δ,t) (or simply Ta) the tree formed from a path Pt on t vertices by attaching Δ-1 P2's on each end of the path Pt, and Tb(Δ, t) (or simply Tb) the tree formed from Pt+2 by attaching Δ-1 P2's on an end of the Pt+2 and Δ-2 P2's on the vertex next to the end. In [X. Li, X. Yao, J. Zhang and I. Gutman, Maximum energy trees with two maximum degree vertices, J. Math. Chem. 45(2009), 962--973], Li et al. proved that among trees of order n with two vertices of maximum degree Δ, the maximal energy tree is either the graph Ta or the graph Tb, where t=n+4-4Δ≥ 3. However, they could not determine which one of Ta and Tb 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 Tb for Δ≥ 7 and any t≥ 3, while the maximal energy tree is Ta for Δ=3 and any t≥ 3. Moreover, for Δ=4, the maximal energy tree is Ta for all t≥ 3 but t=4, for which Tb is the maximal energy tree. For Δ=5, the maximal energy tree is Tb for all t≥ 3 but t is odd and 3≤ t≤ 89, for which Ta is the maximal energy tree. For Δ=6, the maximal energy tree is Tb for all t≥ 3 but t=3,5,7, for which Ta is the maximal energy tree. One can see that for most Δ, Tb is the maximal energy tree, Δ=5 is a turning point, and Δ=3 and 4 are exceptional cases.