New Methods for Calculating the Degree Distance and the Gutman Index

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. Further-more, 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.