On the Maximal Energy Trees with One Maximum and One Second Maximum Degree Vertex

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 d1 >d 2 ≥ 3a ndt ≥ 3, denote by Ta the tree formed from a path Pt on t vertices by attaching d1 − 1 P2’s on one end and d2 − 1 P2’s on the other end of the path Pt ,a ndTb the tree formed fromPt+2 by attaching d1 − 1 P2’s on an end of the Pt+2 and d2 − 2 P2’s on the vertex next to the end. In [14] Yao showed that among trees of order n and two vertices of maximum degree d1 and second maximum degree d2 (d1 >d 2), the maximal energy tree is either the graph Ta or the graph Tb ,w heret = n +4 − 2d1 − 2d2 ≥ 3. However, she 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. We prove that the maximal energy tree is Tb if d1 ≥ 7, d2 ≥ 3o rd1 =6 ,d 2 = 3. Moreover, for d1 = 4 and d2 = 3, the maximal energy tree is the graph Tb if t =4 , and the graph Ta otherwise. For other cases, the maximal energy tree is the graph Ta if (i) d1 =5 ,d 2 =4 ,t is odd and 3 ≤ t ≤ 45, (ii) d1 =5 ,d 2 =3 ,t is odd and 3 ≤ t ≤ 29, (iii) d1 =6 ,d 2 =5 ,t =3 , 5, 7, (iv) d1 =6 ,d 2 =4 ,t = 5; and for all the remaining cases, the maximal energy tree is the graph Tb.