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 $\Delta\geq 3$ and $t\geq 3$, denote by $T_a(\Delta,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices by attaching $\Delta-1$ $P_2$'s on each end of the path $P_t$, and $T_b(\Delta, t)$ (or simply $T_b$) the tree formed from $P_{t+2}$ by attaching $\Delta-1$ $P_2$'s on an end of the $P_{t+2}$ and $\Delta -2$ $P_2$'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 $\Delta$, the maximal energy tree is either the graph $T_a$ or the graph $T_b$, where $t=n+4-4\Delta\geq 3$. 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 $\Delta\geq 7$ and any $t\geq 3$, while the maximal energy tree is $T_a$ for $\Delta=3$ and any $t\geq 3$. Moreover, for $\Delta=4$, the maximal energy tree is $T_a$ for all $t\geq 3$ but $t=4$, for which $T_b$ is the maximal energy tree. For $\Delta=5$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t$ is odd and $3\leq t\leq 89$, for which $T_a$ is the maximal energy tree. For $\Delta=6$, the maximal energy tree is $T_b$ for all $t\geq 3$ but $t=3,5,7$, for which $T_a$ is the maximal energy tree. One can see that for most $\Delta$, $T_b$ is the maximal energy tree, $\Delta=5$ is a turning point, and $\Delta=3$ and 4 are exceptional cases.