  In the paper we develop new methods for calculating the two well-known
topological indices, the degree-distance and the Gutman index. Firstly, we
prove that the Wiener index of a double vertex-weighted graph can be computed
from the Wiener indices of weighted quotient graphs with respect to a partition
of the edge set that is coarser than $\Theta^*$-partition. This result
immediately gives a method for computing the degree-distance of any graph.
Next, we express the degree-distance and the Gutman index of an arbitrary
phenylene by using its hexagonal squeeze and inner dual. In addition, it is
shown how these two indices of a phenylene can be obtained from the four
quotient trees. Furthermore, reduction theorems for the Wiener index of a
double vertex-weighted graph are presented. Finally, a formula for computing
the Gutman index of a partial Hamming graph is obtained.
