The Resistance-Distance Matrix

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) 

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  

Construct the sum-matrix = [L + x O/VH, with x having a nonzero arbitrary value bigger than 0. For the simple graphs, the value of x is taken to be unity. For the weighted graphs, the value of x differs from unity. 

Compute the inverse of the sum-matrix = 1/[L + x O/F]. The inverse is nonsingular for connected graphs. 

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

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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. 

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

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. 

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. 

A square Ax A symmetric matrix, A being the number of non-hydrogen atoms, whose off-diagonal entries are given by the resistance distance Qy between any pair of vertices in the H-depleted molecular graph G as [Klein and Randic, 1993] ... 

The resistance distance between any pair of vertices can also be calculated by the Laplacian matrix as the following ... 

The reciprocal of this matrix, obtained by inverting its off-diagonal elements, is the resistance/distance quotient matrix S2/D (or resistance distance/topological distance quotient matrix), defined as... 

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 ... 

The molecular index based on the resistance-distance/vertex-distance matrix is called the Kirchhoff-sum index (Babic et al 2002). Matrices D/Q and QAD have been used to study the graph cyclicity (Klein and Ivanciuc, 2002) ... 

This equation shows that the resistance-drag force for a certain particle depends on the relative velocities of all the particles of the macromolecule and also on the relative distance between the particles. This expression determines an approximate matrix of hydrodynamic resistance... 

TIMP). Both factors reduce further degradation of connective tissue. The hepatocytes are now multilayered instead of normal single-layered cell plates and lose their microvilh. Fenestration of the sinusoids disappears, whereas the sinusoidal extracellular matrix increases, leading to capillarization of the sinusoids, (s. pp 406, 526) In this way, the distance between the hepatocytes and the blood becomes greater, and the clearance of macromolecular substances is reduced. Stronger flow resistance in the liver leads to portal hypertension. Portoportal and portocentral bands of connective tissue form, in which portosystemic intrahepatic shunts develop. 

Two —> quotient matrices were derived from the resistance matrix [Babic, Klein et al., 2002]. Namely, the distance/resistance quotient matrix D/S2 (or topological distance/resistance distance quotient matrix) was obtained by dividing the off-diagonal elements of the distance matrix D by the corresponding elements of the resistance matrix Q ... 

There is improved abrasion resistance associated with a preferential carbon black-BR phase distribution in blends of NR-BR and SBR-BR. The first abrasion studies on the effects of carbon black phase distribution in NR-BR blends were reported by Krakowski and Tinker (1990a,b). Tread wear resistance was found to increase progressively with increasing carbon black in the BR phase, which was determined from TEM analyses. Tse et al. (1998) have shown that blends of dispersed BIMS in BR matrix failure due to fatigue can be retarded if the mean distance between the crosslinks of the BIMS is less than 60 nm. 

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