The TWKHM conjecture on the number of walks in trees

Let $$w_k(T)$$ be the number of walks of length k in a tree T. Täubig, Weihmann, Kosub, Hemmecke, and Mayr in 2013 proposed the following conjecture (the TWKHM conjecture) that if T is a tree, then for $$k\ge 1$$ , $$\begin{aligned} w_0(T)w_{k+1}(T)-w _1(T)w_{k}(T)\ge 0. \end{aligned}$$ In this paper, using the recurrence relations among the number of walks starting at different vertices, and analyzing the structure of a tree, we prove that the TWKHM conjecture holds if T satisfies one of the following conditions: