Abstract
The new distance defined on a connected graph G contains of three terms: The ordinary distance between any two vertices in G, both the sum and the product of the two vertices' degrees, as this distance is more useful than the ordinary distance, especially in chemical structures because of its effect on the number of bonds (edges) on the atoms (vertices) carbon (graph). In this article, were found index with respect to new distance (d- index) of regular graph, in addition, finding the relationships between d-index. Also, The relationships and The graph were found between the diameter and the radius for the ordinary distance and between the diameter and the radius for the new distance, and finally, The d-index was found for the join operation of two and nth graphs.
