Comparison of mean hitting times for a degree-biased random walk

Consider the random walk on graphs such that, at each step, the next visited vertex is a neighbor of the current vertex, chosen with probability proportional to the inverse of the square root of its degree. On one hand, for every graph with n vertices, the maximal mean hitting time for this degree-biased random walk is asymptotically dominated by n. On the other hand, the maximal mean hitting time for the simple random walk is asymptotically dominated by n. Yet, in this article, we exhibit for each positive integer n: • A graph of size n with maximal mean hitting time strictly smaller for the simple random walk than for the degree-biased one. • A graph of size n with mean hitting time of a so called root vertex strictly smaller for the simple random walk than for the degree-biased one.