eR(T2)=
0
3.31
0
0
0
0
0
0
0
3.83
0
0
0
0
3.31
0
3.68
0
0
3.27
0
0
3.64
0
0
0
0
3.18
0
0
0
0
0
0
0
0
The first numerical matrix, when the connectivity index is employed, is named the edge-Randić matrix and denoted by eR. An example of this matrix obtained from the edge-graphical matrix of T2 is given below. For practical reasons only the upper triangle of the matrix is shown.
eR(T2)= |
|
0 |
3.31 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
3.83 |
0 |
0 |
0 |
0 |
3.31 |
||||
0 |
3.68 |
0 |
0 |
3.27 |
0 |
|||||
0 |
3.64 |
0 |
0 |
0 |
||||||
0 |
3.18 |
0 |
0 |
|||||||
0 |
0 |
0 |
||||||||
0 |
0 |
|||||||||
0 |
From this matrix can be derived, for example, the vertex-Randić-Wiener index, obtained by summing the elements of the matrix-triangle.
The second numerical matrix is named the path-Randić matrix and denoted by pR. An example of this matrix obtained from the path-graphical matrix of T2 is given below and again only the upper triangle of the matrix is shown.
pR(T2)= |
|
0 |
3.31 |
3.41 |
3.00 |
3.00 |
2.00 |
2.91 |
2.77 |
|
0 |
3.83 |
3.83 |
3.41 |
2.41 |
3.33 |
3.31 |
||||
0 |
3.68 |
3.27 |
2.27 |
3.27 |
3.41 |
|||||
0 |
3.64 |
2.64 |
2.82 |
3.41 |
||||||
0 |
3.18 |
2.41 |
3.00 |
|||||||
0 |
1.41 |
2.00 |
||||||||
0 |
2.91 |
|||||||||
0 |
From this matrix can be obtained, for example, the path-Randić-Wiener index by summing up the elements in the matrix-triangle.
The third numerical matrix is named the sparse vertex-Randić matrix and denoted by svR. An example of this matrix obtained from the sparse vertex-graphical matrix of T2 is given below. Only the upper matrix-triangle is given.
svR(T2)= |
|
0 |
2.41 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
1.41 |
0 |
0 |
0 |
0 |
2.41 |
||||
0 |
2.41 |
0 |
0 |
2.83 |
0 |
|||||
0 |
2.27 |
0 |
0 |
0 |
||||||
0 |
2.64 |
0 |
0 |
|||||||
0 |
0 |
0 |
||||||||
0 |
0 |
|||||||||
0 |
The summation of the matrix-elements in the above matrix-triangle gives the sparse vertex-Randić-Wiener index.
The fourth numerical matrix is named the dense vertex-Randić matrix and denoted by dvR. An example of this matrix obtained from the dense vertex-graphical matrix of T2 is given below and again only the upper matrix-triangle is given.
dvR(T2)= |
|
0 |
2.41 |
2.41 |
2.91 |
2.27 |
2.81 |
2.91 |
2.77 |
|
0 |
1.41 |
2.00 |
1.41 |
1.91 |
1.91 |
2.41 |
||||
0 |
2.41 |
1.41 |
2.41 |
2.83 |
2.41 |
|||||
0 |
2.27 |
2.27 |
2.41 |
2.91 |
||||||
0 |
2.64 |
2.27 |
2.27 |
|||||||
0 |
2.77 |
2.81 |
||||||||
0 |
2.91 |
|||||||||
0 |
The molecular descriptor based on this matrix, obtained by summing the elements in its triangle, is called the dense vertex-Randić-Wiener index.
