[gD/tD]ij= 
The distance/distance matrix, denoted by D/D, have been introduced by Randić et al. [280] and briefly discussed by Todeschini and Consonni [21]. These matrices are defined in terms of geometric or topographic distances g(i,j) and the graph-theoretical (topological) distances l(i,j). Thus, they unify the topological and topographic (geometric) information on the structure of a given molecule.
We consider here two kinds of the distance/distance matrices: the topographic distance/topological distance matrix and the corresponding reciprocal matrix. We denote the first matrix as gD/tD and the second as tD/gD.
The topographic distance/topological distance matrix is an unsymmetric V × V matrix, in which the upper matrix-triangle represent the same part of the corresponding topographic matrix and the lower matrix-triangle the same part of vertex-distance matrix (defined in section 4.1):
[gD/tD]ij= |
g (i,j) if i < j |
| 0 if i = j | |
| l (i,j) if i >j (83) |
The topographic matrix, denoted by TM, is defined as [281]:
[TM]ij= |
g(i,j) if i ≠ j |
| 0 otherwise (84) |
The geometric distance matrices are based on the actual molecular geometry of a molecule in 3-D space [282,283,284] while the topographic matrices are based on standardized bond angles and bond lengths. They can be derived by embedding a graph on a regular 2-D or 3-D grid. The embedding of T2 on a graphite (honeycomb) grid is shown in Figure 40.
The corresponding topographic matrix is given by:
TM(T2)= |
 |
0 |
1 |
√3 |
√7 |
2√3 |
√19 |
2 |
√3 |
 |
1 |
0 |
1 |
√3 |
√7 |
2√3 |
√3 |
1 |
|||
√3 |
1 |
0 |
1 |
√3 |
√7 |
1 |
√3 |
|||
√7 |
√3 |
1 |
0 |
1 |
√3 |
√3 |
2 |
|||
2√3 |
√7 |
√3 |
1 |
0 |
1 |
2 |
3 |
|||
√19 |
2√3 |
√7 |
√3 |
1 |
0 |
3 |
√13 |
|||
2 |
√3 |
1 |
√3 |
2 |
3 |
0 |
√7 |
|||
√3 |
1 |
√3 |
2 |
3 |
√13 |
√7 |
0 |
The elements of this matrix are computed by taking the edge-distance to be unity and using the plane geometry. The topographic matrices have been used to produce the topographic invariants for the structure-property-activity studies for 3-D structures.
It should be noted that the structures of topographic matrices depend on the embedding – different embeddings of the same graph result in different matrices. As a result, structural invariants derived from such matrices depend on the assumed 'conformation' of a graph [281] .
Todeschini and Consonni [21] also presented the reciprocal topographic (geometric) distance matrix, denoted by TM-1,defined as:
[TM-1]ij= |
1/g(i,j) if i ≠ j |
| 0 otherwise (85) |
The TM-1 matrix of T2 embbed on a graphite grid is given below.
TM-1(T2)= |
|
0 |
1 |
1/√3 |
1/√7 |
1/2√3 |
1/√19 |
1/2 |
1/√3 |
|
1 |
0 |
1 |
1/√3 |
1/√7 |
1/2√3 |
1/√3 |
1 |
|||
1/√3 |
1 |
0 |
1 |
1/√3 |
1/√7 |
1 |
1/√3 |
|||
1/√7 |
1/√3 |
1 |
0 |
1 |
1/√3 |
1/√3 |
1/2 |
|||
1/2√3 |
1/√7 |
1/√3 |
1 |
0 |
1 |
1/2 |
1/3 |
|||
1/√19 |
1/2√3 |
1/√7 |
1/√3 |
1 |
0 |
1/3 |
1/√13 |
|||
1/2 |
1/√3 |
1 |
1/√3 |
1/2 |
1/3 |
0 |
1/√7 |
|||
1/√3 |
1 |
1/√3 |
1/2 |
1/3 |
1/√13 |
1/√7 |
0 |
The topographic distance/topological distance matrix of T2 is as follows:
| gD/tD(T2)= | |
0 |
1 |
√3 |
√7 |
2√3 |
√19 |
2 |
√3 |
|
1 |
0 |
1 |
√3 |
√7 |
2√3 |
√3 |
1 |
|||
2 |
1 |
0 |
1 |
√3 |
√7 |
1 |
√3 |
|||
3 |
2 |
1 |
0 |
1 |
√3 |
√3 |
2 |
|||
4 |
3 |
2 |
1 |
0 |
1 |
2 |
3 |
|||
5 |
4 |
3 |
2 |
1 |
0 |
3 |
√13 |
|||
3 |
2 |
1 |
2 |
3 |
4 |
0 |
√7 |
|||
2 |
1 |
2 |
2 |
4 |
5 |
3 |
0 |
The normalized Perron root (the first eigenvalue) [285] of such matrices for linear structures appears to be an index of molecular folding [286,287].
The reciprocal matrix of gD/tD(T2), that is, the topological distance/topographic distance matrix of T2 is:
tD/gD(T2)= |
|
0 |
1 |
2 |
3 |
4 |
5 |
3 |
2 |
|
1 |
0 |
1 |
2 |
3 |
4 |
2 |
1 |
|||
√3 |
1 |
0 |
1 |
2 |
3 |
1 |
2 |
|||
√7 |
√3 |
1 |
0 |
1 |
2 |
2 |
3 |
|||
2√3 |
√7 |
√3 |
1 |
0 |
1 |
3 |
4 |
|||
√19 |
2√3 |
√7 |
√3 |
1 |
0 |
4 |
5 |
|||
2 |
√3 |
1 |
√3 |
2 |
3 |
0 |
3 |
|||
√3 |
1 |
√3 |
2 |
3 |
√13 |
√7 |
0 |
A number of graph invariants can be obtained from the distance/distance matrices made up from geometric, topographic and topological matrices [21].
