Resistance distance matrix

Babic, D., Klein, D.J., Lukovits, 1., Nikolic, S. and Trinajstic, N. (2002) Resistance-distance matrix a computational algorithm and its application. Int. J. Quant. Chem., 90, 166-176. 

Ivanciuc, O. (2002d) Design of topological indices. Part 29. QSAR and QSPR structural descriptors from the resistance distance matrix. Rev. Roum. Chim., 47, 675-686. 

The resistance-distance matrix of a vertex-labeled connected graph G, denoted by Q, is a real symmetric VxV matrix defined as (Klein and Randid, 1993)... 

As an example, the resistance-distance matrix of the vertex-labeled graph Gj (see structure A in Figure 2.1) is given below ... 

An algorithm based on the Laplacian matrix has been proposed for efficacious computing of the resistance-distance matrix for connected graphs (Babid et al., 2002). This computational algorithm consists of the following steps ... 

Compute the resistance-distance matrix Q using the elements of the ... 

Application of the Algorithm for Computing the Resistance-Distance Matrix of a Simple Graph... 

The resistance-distance matrix 12 of Gj (see this matrix above)... 

The Wiener-like distance index, named the Kirchhoff index (Bonchev et al., 1994 Gutman et al., 2003 Zhou and Trinajstic, 2008, 2009b), is based on the resistance-distance matrix. However, it has been elegantly demonstrated (Gutman and Mohar, 1996) that the quasi-Wiener index (Mohar et al., 1993 Gutman et al., 1994 Markovid et al., 1995) and the Kirchhoff index are identical topological indices. 

D. Babic, D.J. Klein, I. Lukovits, S. Nikolic, and N. Trinajstid, Resistance-distance matrix A computional algorithm and its apphcation, Int. J. Quantum Chem. 90 (2002) 166-176. A.T. Balaban, Topological indices based on topological distances in molecular graphs. Pure Appl. Chem. 55 (1983) 199-206. 

S. Nikolid, N. Trinajstid, and B. Zhou, On the eigenvalues of the ordinary and reciprocal resistance-distance matrix, in Computational methods in science and engineering, Vol. I, ed. G. Marouhs and T.E. Simos, American Institute of Physics, Melville, NY, 2009, pp. 205-214. 

Several quotient matrices are in use. Here we list six the vertex-distance/detour matrix D/DM (Randid, 1994b), the detour/vertex-distance matrix DMAD (Plavsid et al., 1998), the vertex-distance/resistance-distance matrix D/Q (Babid et al., 2002 Klein and Ivanciuc, 2002), the resistance-distance/vertex-distance matrix 12AD (Babid et al., 2002 Klein and Ivanciuc, 2002), the vertex-distance/vertex-distance-complement matrix DA D (Nikolid et al., 2001a), and the vertex-distance-complement/vertex-distance matrix " DAD (Nikolid et al, 2001a). These six quotient matrices for Gj (see structure A in Figure 2.1) are given below ... 

If one assumes that all bonds have the same (unit) resistance one can write a resistance-distance matrix 0. This matrix has also been referred to as a Kirchhoff matrix, in view of the fact that it rests on Kirchhoff s current flow laws. The resistance-distance matrix better reflects interatomic distances in cyclic compounds than the ordinary distance matrix, as it takes into account not only the shortest paths in a graph but also the presence of alternative connections between vertices. 

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