The Complementary Vertex-Distance Matrix

The complementary vertex-distance matrix, denoted by D, has been introduced by Ivanciuc (2000c) and discussed by Balaban et al. (2000) and Ivanciuc et al. (2000). It is a square symmetric VxV matrix defined as 

The diameter D of a graph G is the longest geodesic distance between any two vertices i and j in G, i.e., the largest [ D]y value in the vertex-distance matrix (Harary, 1971). 

The elements of the complementary vertex-distance matrix differ from the elements of the reverse-Wiener matrix only for unity (see Section 5.5). 

The complementary vertex-distance matrices are used to generate the Wienerlike molecular descriptors that have been successfully tested in QSPR modeling (Ivanciuc et al., 2000). The complementary vertex-distance matrices of vertex- and edge-weighted graphs have also been introduced and used in QSPR (Ivanciuc, 2000c). 

The reciprocal of the complementary vertex-distance matrix, denoted by is simply given by 

---

Ivanciuc (2000c) extended the concept of reciprocal of the complementary vertex-distance matrix to the vertex- and edge-weighted graphs and used the derived Wiener-like indices in QSPR modeling. 

For vertex- and edge-weighted molecular graphs, the complementary distance matrix was defined as [Ivanciuc, 2000i]... 

---