A Relation between D-Index and Wiener Index for r-Regular Graphs

For any two distinct vertices and in a connected graph , let be the length of path and the D–distance between and of is defined as: , where the minimum is taken over all paths and the sum is taken over all vertices of path . The D-index of G is defined as . In this paper, we found a general formula that links the Wiener index with D-index of a regular graph G. Moreover, we obtained different formulas of many special irregular graphs