A Metric Graph for Which the Number of Possible Endpoints of a Random Walk Grows Minimally

The paper proves that, for incommensurable edge lengths, graphs with eventually minimal growth of NΓ(T) are exactly unions of chains attached to the root and that each chain has length at least 2, via an asymptotic expansion N(T)=N1 T^{|E|-1}+N2 T^{|E|-2}+o(T^{|E|-2}) and two surgeries (T1, T2) that strictly decrease the second-term coefficient P2 among trees . The candidate solution reaches the same conclusion by a different, direct dynamical argument that exploits canonical singleton return times 2ns (guaranteed by Q-linear independence) to show that cycles and non-root branching, as well as length-1 chains at the root, can be removed by local surgeries that reduce NΓ(T) by a quantity growing linearly in T. Aside from minor clarifications (e.g., handling the |E|=1 degenerate case explicitly), both arguments are coherent and consistent with the formal model from the paper’s introduction .