Is this still a vector space?

Let us define the product of two matrices $a \times b = c$ as follows : (in terms of the element at the $i$ th row $j$ th column).

$c_{ij} = \min( a_{ik} + b_{kj} ) \ \ \ 1 \leq k \leq n$

Now suppose we are given the adjacency matrix $A$ of some directedgraph $G$ . Using our definition of multiplication above, what would $(A^5)_{34}$ represent? (Suppose there exists a path between node $3$ and $4$ )