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

New methods for calculating the degree distance and the Gutman index