Topological Indices Based on the Distance Matrix

The distance matrix D(G) of a graph G is another important graph-invariant. Its entries dy, called distances, are equal to the number of edges connecting the vertices i and j on the shortest path between them. Thus, all dy are integers, including du = 1 for nearest neighbours, and, by definition, d = 0. The distance matrix can be derived readily from the adjacency matrix  

Here Bj(G) stands for matrices containing as single non-zero entries b2 = 2, b3 = 3 etc. For example, B2 contains the shortest paths between the second neighbours, B3 between the third neighbours etc. 

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