Find the k length path that maximize a node associate metric
A brute-force approach is to do a backtracking search starting from $s$.  Something like:
Search($s$, $n$, $S$):

If $n=1$, yield $S \cup \{s\}$ and return.  Otherwise:

For each vertex $v$ that is adjacent to $s$, do:
a. Call Search($v$, $n-1$, $S \cup \{s\}$).


If you invoke Search($vs$, $10$, $\emptyset$), this will generate all possible sets of vertices that can arise from a path of length 10; now calculate the cost of each such subset, and keep the largest one.
$6^{10}$ is about 60 million, so this might complete within a few seconds on most graphs.