Optimum addition of an edge to a path

Let
Q(G)
denote the sum taken over all vertex pairs
u,v
of a graph
G
of the shortest distance from
u
to
v
in
G
. We determine the edges e which when added to the path
P_m
minimize
Q(P_{m} + e)
. For example, if
1,{\cdots},2n+ 1
is the vertex sequence of
P_{2n +1}
, the unique optimum edge is
e=[a, 2n+2-a]
, where
a = [n(\sqrt{2} - 1)]
.