Wiener Matrices

JJ indices derived from the - Wiener matrix were proposed as a generalization of the Balaban index in analogy with the Kier-Hall -> connectivity indices. 

It is worth pointing out that the symmetric Cluj-distance matrix CJD is identical to the Wiener matrix W for acyclic graphs obviously, the corresponding - graph invariants coincide. 

For acyclic graphs, the matrix CJA is equal to the symmetric Cluj-distance matrix CJD and therefore to the Wiener matrix W, while it is different from the -> Szeged matrix SZ. 

For acyclic graphs, the hyper-detour index ww is equal to the index Dp obtained from the distance-path matrix Dp and to the WW obtained from the - Wiener matrix W. 

For acyclic graphs the hyper-distance-path index Dp coincides with the - hyper-Wiener index WW derived from the - Wiener matrix and with the -> hyper-detour index derived from the detour-path matrix. Moreover, it was proposed as an extension of the hyper-Wiener index for any graph [Klein et al., 1995],... 

Each entry (a subgraph) of the GG matrix can be transformed into a matrix which is a submatrix of the GG matrix and any topological matrix (e.g. - adjacency matrix A, -> distance matrix D, -> Wiener matrix W, - distance/distance matrix DD, - distance/ detour quotient matrix D/A, etc) can be chosen to represent molecular subgraphs. 

They are the eigenvalues obtained from the Wiener matrix W. In particular, in analogy with the Lovasz-Pelikan index, the 1 Wieuer matrix eigenvalue X] (or Wiener matrix leading eigenvalue) is taken as an alternative descriptor for the molecular... 

The Harary index and hyper-Harary index, defined only for acyclic graphs, are obtained from, respectively, the P -order sparse -> reciprocal Wiener matrix W ... 

It must be noted that, while the indices obtained by applying the Wiener operator to the distance matrix D and to the P -order - sparse Wiener matrix are equal (i.e. the Wiener index), the corresponding Harary indices are not, i.e. H... 

The simplest form to represent the chemical information contained in a molecular graph is by a -> matrix representation of a molecular structure. Examples are -+ adjacency matrix A, - edge adjacency matrix E, vertex - distance matrix D, -> edge distance matrix D, - incidence matrix I, - Wiener matrix W, -> Hosoya Z-matrix Z, - Cluj matrices CJ, - detour matrix A, - Szeged matrix SZ, -> distance/distance matrix DD, and - detour/distance matrix A/D. 

Wiener delta matrix -> Wiener matrix Wiener index Wl ( Wiener number)... 

For acyclic graphs the Wiener index, according to its original definition, can also be obtained from the - Wiener matrix W by summing all of the entries for paths i-j of length one Vy (i- - bonds b) above the main diagonal ... 

The /th row sum of the Wiener matrix is called the Wiener matrix degree g, ... 

Among the properties of the Wiener matrix, it can be noted that (a) entries corresponding to paths between - terminal vertices are necessarily equal to 1 (b) rows of the matrix corresponding to terminal vertices all have entries less than A (the number of vertices) (c) an entry with value A - 1) shows a terminal bond, because only terminal bonds divide the vertices into 1 + (A - 1) partition (d) terminal vertices have a smaller row sum (Wiener matrix degree) than the centrally located ones associated with larger row sums. 

The Wiener matrix for acyclic graphs can also be obtained by the -+ Cluj-distance matrix CJDu as ... 

The sparse Wiener matrix of m th order W is derived from the Wiener matrix setting to zero all entries except those corresponding to m th order paths "py. This matrix can be calculated by the Hadamard matrix product of the Wiener matrix and the -> binary sparse matrix B whose elements corresponding to m th order paths are equal to one, or else zero ... 

Several Wiener matrix invariants [Randic et al, 1994a] can be calculated. 

In analogy with the Kier-Hall - connectivity indices x and the Balaban distance connectivity index J, JJ indices [Randic et al, 1994a] are derived from the Wiener matrix based on the Wiener matrix degrees q, ... 

The eigenvalues of the Wiener matrix are among -+ eigenvalue-based descriptors. The reciprocal Wiener matrix W is a matrix whose elements are the reciprocal of the corresponding Wiener matrix elements ... 

Wiener matrix eigenvalues -> eigenvalue-based descriptors... 

Wiener matrix leading eigenvalue Wiener matrix eigenvalue -> eigenvalue-based descriptors (O Wiener matrix eigenvalues)... 

Diudea, M.V. and Pop, C.M. (1996). Molecular Topology. 27. A Schultz-Type Index Based on the Wiener Matrix. Indian J.Chem., 35A, 257-261. 

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