Distance-Weighted Adjacency Matrix

Randid and his coworkers (Randid et al., 2010) introduced the distance-weighted adjacency matrix, denoted by DWA, as a part of their project on developing the natural distance matrices. The DWA matrix for acyclic graphs is defined as 

By substituting [d(i) + d(j)V with [d i) + dO)l in (2.55), oue obtains the sum-vertex-counectivity matrix (Zhou and Trinajstid, 2009, 2010a Randid et al., 2010) see Section 2.15. If one substitutes [d(i) + d(j)f with either [d(i) + d(jj or [d(i) + dO)] the sum-edge-Zagreb matrix or modified-sum-edge-Zagreb matrix is obtained, respectively. Thus, it appears that all these matrices can be traced to the Randid connectivity matrix proposed years ago (Randid, 1992). 

Aringhieri, P. Hansen, and F. MaluceUi, Chemical trees enumeration algorithms, Universita di Paoia, Dipartimento di Informatice, Pisa, 1999. 

Balandin, Structural algebra in chemistry, Acta Physicochim. USSR 12 (1940) 447-479. K. Balasubramanian, Apphcations of edge-colorings of graphs and characteristic polynomials to spectroscopy and quantum chemistry, J. Mol. Struct. (Theochem) 185 (1989) 229-248. 

Barysz, D. Plavsic, and N. Trinajstic, A note on topological indices, MATCH Commun. Math, Comput. Chem, 19 (1986) 89—116. 

---

The determinant of (A" -i- G) was also proposed as a molecular topographic descriptor where A is the weighted adjacency matrix (- weighted matrices) where the entries corresponding to bonded atoms are - bond distances and G the geometry matrix [Mihalic et ai, 1992a]. 

The sum matrix resulting from the geometry matrix G and the bond length-weighted adjacency matrix A, where elements corresponding to the pairs of adjacent vertices are bond distances, was defined by Mihalic et al. [Mihalic, Nikolic et al., 1992]. Its determinant, that is, the —> def A + G index, and —> 3D-SchuIfzinde%wereproposedandusedinQSARmodelingas the molecular descriptors. 

A bond distance-weighted edge adjacency matrix is derived from the edge adjacency matrix using - bond distances calculated by computational chemistry methods as diagonal entries on selected molecular geometries [Estrada, 1997] ... 

As for the edge adjacency matrix, the spectral moments of the bond distance-weighted edge adjacency matrix E were defined [Estrada, 1997 Estrada, 1998a] as ... 

Edge invariants can also be directly obtained by physico-chemical properties of the bonds used as the —> weighting scheme for graph edges, such as bond dipole moments, bond order indices, and so on, as well as from the —> edge adjacency matrix and edge distance matrix by applying specific matrix operators such as the row sum operator. 

Most of the graph-theoretical matrices are symmetrical, whereas some of them are un-symmetrical. Examples of unsymmetrical matrices are —> Szeged matrices, —> Cluj matrices, random walk Markov matrix, —> combined matrices such as the topological distance-detour distance combined matrix, and some weighted adjacency and distance matrices. 

E(r) Bond distance-weighted edge adjacency matrix B B s... 

From Ivanciuc-weighted distance matrices, the corresponding Ivanduc weighted adjacency matrices are derived by setting equal to zero all the matrix entries corresponding to pairs of nonadjacent vertices ... 

The usage of matrix representatives of molecules is not restricted to LDMs and can be extended to quantities directly derivable from both theory and experiment such as the matrix of Coulombic nuclear-nuclear repulsion, the distance matrix, or the matrices of bond critical point (BCP) properties such as the electron density-weighted adjacency/connectivity matrix (EDWAM/EDWCM) [22-24, 30]. 

---