On the maximal energy tree with two maximum degree vertices

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 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 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 one exception that t=4, for which Tb is the maximal energy tree.For Δ=5, the maximal energy tree is Tb for all t⩾3 but 44 exceptions that t is both 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 three exceptions that t=3,5,7, for which Ta is the maximal energy tree.One can see that for most cases of Δ, Tb 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, Ta has maximal energy, with only one exception Ta(4,4).