The reciprocal complementary Wiener number of a connected graph
G is defined as
where
is the vertex set.
is the distance between vertices
u and
v, and
d is the diameter of
G. A tree is known as a caterpillar if the removal of all pendant vertices makes it as a path. Otherwise, it is called a non-caterpillar. Among all
n-vertex non-cater- pillars with given diameter
d, we obtain the unique tree with minimum reciprocal complementary Wiener number, where
. We also determine the
n-vertex non-caterpillars with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers.