  Given a graph $G$, we denote by $f(G,u_0,k)$ the number of paths of length
$k$ in $G$ starting from $u_0$. In graphs of maximum degree 3, with edge
weights $i.i.d.$ with $exp(1)$, we provide a simple proof showing that (under
the assumption that $f(G,u_0,k)=\omega(1)$) the expected weight of the heaviest
path of length $k$ in $G$ starting from $u_0$ is at least
  \begin{align*}
  (1-o(1))\left(k+\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right),
  \end{align*}
  and the expected weight of the lightest path of length $k$ in $G$ starting
from $u_0$ is at most
  \begin{align*}
  (1+o(1))\left(k-\frac{\log_2\left(f(G,u_0,k)\right)}{2}\right).
  \end{align*}
  We demonstrate the immediate implication of this result for Hamilton paths
and Hamilton cycles in random cubic graphs, where we show that typically there
exist paths and cycles of such weight as well. Finally, we discuss the
connection of this result to the question of a longest cycle in the giant
component of supercritical $G(n,p)$.
