Ordering trees by the minimal entries of doubly stochastic graph matrices

Given an n-vertex graph G, the matrix Ω(G) = (In + L(G))−1 = (ωij) is called the doubly stochastic graph matrix of G, where In is the n × n identity matrix and L(G) is the Laplacian matrix of G. Let ω(G) be the smallest element of Ω(G). Zhang and Wu [X.D. Zhang and J.X. Wu. Doubly stochastic matrices of trees. Appl. Math. Lett., 18:339–343, 2005.] determined the tree T with the minimum ω(T ) among all the n-vertex trees. In this paper, as a continuance of the Zhang and Wu’s work, we determine the first ⌈ 2 ⌉ trees T1, T2, . . . , T⌈n−1 2 ⌉ such that ω(T1) < ω(T2) < · · · < ω (