Shortest-weight paths in random regular graphs

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hatα\log n$ edges where $\hatα$ is the unique solution of the equation $α\log(\frac{d-2}{d-1}α) - α= \frac{d-3}{d-2}$, for $α> \frac{d-1}{d-2}$.

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