Path counts

Table 2. Path counts in a cubic lattice as a function of path length between two points ten links apart...

This 3D lattice Hamiltonian with two traps can be elaborated further by adding filled lattice orbitals (3D bridge orbitals), which will then exhibit interference effects, and which could be computed by an extension of the Hopfield-Beratan method. Nevertheless the path counting given in Table 2 serves to illustrate the dramatic character of the difference between the single path model (Eq. 15) and the many path result (Eq. 14) in three dimensions. 

It is clear that the evaluation of moments from closed path counting becomes cumbersome as n increases the higher moments are evidently dominated by walks weaving around the central atom. This is the basic reason for the well-known ill-conditioned nature of the problem. 

The first proposed graphical bond order was calculated using the -> total path count P as a molecular invariant and is therefore denoted by the ratio (P / P)y, where i and j refer to the vertices incident to the bth bond erased from the graph [Randic, 1991b]. 

The L index was proposed as the molecular descriptor defined as the simple linear combination of molecular path counts of order one P (the number of bonds), order two (the -> connection number Na), and order three P ... 

Moreover, path count differences P - P and P - P [Randic and Trinajstic, 1988] and connectivity differences x X nd x / re usually encountered in QSAR modeling the following path count combination P + P + P2 — P3 was also found as the critical parameter in the correlation of carbon-13 chemical shift sums in alkanes [Miyashita et al., 1989]. 

These are simple molecular descriptors based on counting the defined elements of a compound. The most common chemical count descriptors are atom number A, bond number B, cyclomatic number C, -> hydrogen-bond acceptor number and -> hydrogen-bond donor number, -> distance-counting descriptors, -> path counts, walk counts. 

The first element of each SE-vector, i.e. the 0-order molecular path count, corresponds to the count of the considered graph elements for example, the first entry in the TT-vector is the number of atoms in the molecule, in the OT-vector it is the number of oxygen atoms, and in the 2T-vector the number of double bonds. 

Molecular BO numbers (MID) are molecular descriptors defined as - weighted path counts or weighted - walk counts, mainly proposed to uniquivocally identify a molecule by a single real number, the aim being to obtain highly discriminatory power [Randic, 1984b Szymanski et al, 1986a],... 

It is the molecular identification number proposed first [Randic, 1984b Szymanski et al, 1985 Randic and Trinajstic, 1993a] and is defined as a weighted molecular path count ... 

This ID number accounts for multiple bonds in the molecule for saturated molecules each bond weight is equal to one, therefore the ID number coincides with the - total path count. 

It is analogously defined as a weighted molecular path count ... 

From the geometry matrix, the usual -> graph invariants can be calculated such as -t characteristic polynomial, -> eigenvalue-based descriptors, -> path counts, - ID numbers, -> 3D-Balaban index, -> 3D-Schultz index and so forth [Randic, 1988b Nikolic et al, 1991]. It is noteworthy that all these indices are sensitive to molecular geometry. Moreover, the geometry matrix is used for the calculation of size descriptors and - 3D-MoRSE descriptors. 

The vertex path code (or Ran c atomic path code) of the ith vertex is the ordered sequence of atomic path counts, with respect to the path length ... 

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