The Path Matrix

The path matrix, denoted by P, has been introduced by Randid (1991a Randid and Trinajstid, 1993) and used by Randid for structural ordering and branching of acyclic saturated hydrocarbons (Randid, 1997a, 1997b, 1998). 

The P matrix of a vertex-labeled connected simple graph G is a square V x V matrix whose entries are defined as follows  

FIGURE 5.1 Entries to the path matrices of Tj (a) and G, (b). The broken lines represent the removed edges. 

The quantity p i,j)lp could be considered a graphical bond order of the edge i-j in G (Randid et al., 1997). By summing up the nonvanishing entries in the upper (or the lower) matrix-triangle, the P lP index is obtained (Randid, 1991b). It has also been 

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This is the sum of the positive eigenvalues A, of the path-% matrix Xp, based on the path connectivities calculated by the valence vertex degree 5 of the atoms in the path [Sagan and Inel, 1993, 1995 Sagan and Balcioglu, 1996] ... 

For aqrclic graphs, connectivity indices for path subgraphs can be calculated by the Wiener operator applied to the product of the path-% matrix and the geodesic matrix B whose elements are all equal to zero except for those corresponding to the shortest paths i-j of length k that are equal to 1 ... 

A variant of the path-% matrix is obtained by using the vertex distance sum a in place of the vertex degree as the vertex-weighting scheme. The matrix obtained in this way can be considered an extension of the —> distance-sum-connectivity matrix and thus called path-distance-sum-connectivity matrix, denoted by "Xy... 

The degree of vertex is equivalent to the number of edges. Equation (1) describes the vertex-to-vertex adjacency. It is an Ny X Ny symmetric matrix having zero diagonal elements. Equation (2), however, defines incidence matrix that outlines vertices and edges. Lastly, equation (3) defines the path matrix that stores information about all paths that emanate from the root. It is a -1) matrix excluding... 

In this section, we insert explicitly. The sum has the following meaning for each pair of points x and a draw a path j (t) that starts at x (j (t = 0) = a ) and ends at x" (j (t = r) = a ). This path need not be a classical patii (see figure B3.4.15). Each such path contributes c where S is the action of the path. S (iimelated to the scattering matrix) is calculated very simply as... 

Note the meaning of this expression for each choice of the initial and final position a and a , calculate the classical path that takes you from x to x" m time t. Specifically, calculate tire momentum along the path and the final momentum, p", and find out how p" varies with the initial position. This would give, for a multidimensional problem, a matrix dp"-Jdx"- whose absolute detenninant needs to be inverted. 

Thus B is a diagonal mati ix that contains in its diagonal (complex) numbers whose norm is 1 (this derivation holds as long as the adiabatic potentials are nondegenerate along the path T). From Eq. (31), we obtain that the B-matrix hansfomis the A-matrix from its initial value to its final value while tracing a closed contour ... 

The distance matrix D of a graph G with n vertices is a square n x n symmetric matrix as represented by Eq. (13), where is the distance between the vertices Vi and Vj in the graph (i.e., the number of edges on the shortest path). 

In graph theory, the conversion of the adjacency matrix into the distance matrix is known as the "all pairs shortest path problem",... 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

Directed Oxidation of a Molten Metal. Directed oxidation of a molten metal or the Lanxide process (45,68,91) involves the reaction of a molten metal with a gaseous oxidant, eg, A1 with O2 in air, to form a porous three-dimensional oxide that grows outward from the metal/ceramic surface. The process proceeds via capillary action as the molten metal wicks into open pore channels in the oxide scale growth. Reinforced ceramic matrix composites can be formed by positioning inert filler materials, eg, fibers, whiskers, and/or particulates, in the path of the oxide scale growth. The resultant composite is comprised of both interconnected metal and ceramic. Typically 5—30 vol % metal remains after processing. The composite product maintains many of the desirable properties of a ceramic however, the presence of the metal serves to increase the fracture toughness of the composite. 

Example 4. For a given lattice, a relationship is to be found between the lattice resistivity and temperature usiag the foUowiag variables mean free path F, the mass of electron Af, particle density A/, charge Planck s constant Boltzmann constant temperature 9, velocity and resistivity p. Suppose that length /, mass m time /, charge and temperature T are chosen as the reference dimensions. The dimensional matrix D of the variables is given by (eq. 55) ... 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

This is a crucial point because (as we will see) proton transport is coupled with ATP synthesis. Oxidation of one FADHg in the electron transport chain results in synthesis of approximately two molecules of ATP, compared with the approximately three ATPs produced by the oxidation of one NADH. Other enzymes can also supply electrons to UQ, including mitochondrial 5w-glyc-erophosphate dehydrogenase, an inner membrane-bound shuttle enzyme, and the fatty acyl-CoA dehydrogenases, three soluble matrix enzymes involved in fatty acid oxidation (Figure 21.7 also see Chapter 24). The path of electrons from succinate to UQ is shown in Figure 21.8. 

If the edge jump is too large, the sample should be diluted or the path-length decreased. If the edge jump is too small, then addition of more sample is one possibility, although this is dependent on the matrix in which the sample is studied. 

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