2 problems

Let $P_n$ be the path graph on $n$ vertices. For a weight vector $x$ on $P_n$, define … where $\overline{x}$ denotes the average-weight vector. Path-graph averaging conjecture. For…

Let $S_n$ be the star graph on $n+1$ vertices, and let $x$ be the weight vector on $S_n$ that gives weight $1-\frac{1}{n+1}$ to the central vertex and weight $-\frac{1}{n+1}$ to ev…