The Distance Polynomial

Analogously to the characteristic polynomial derived from adjacency matrix A, Hosoya et al. 64 introduced the distance polynomial  

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The distance polynomial is the characteristic polynomial of the distance matrix D of the molecular graph [Hosoya, Murakami et al, 1973 Graham, Hoffman et al, 1977 Graham and Lovasz, 1978] ... 

As for the distance polynomial, the coefficients other than Cg, which is always equal to one, are negative. 

Note that, 2-methylpentane being an acyclic molecule, the detour polynomial coincides with the distance polynomial. 

The same approach applied to the distance polynomial led to the definition of the Hosoya Z index (or Z index) [Hosoya, Murakami et al., 1973] ... 

The distance polynomial of the molecular graph G is the characteristic polynomial of its distance matrix D(G) " " ... 

The distance spectrum of the graph G, Sp(D,C), is the set of eigenvalues of the distance matrix D(C), or the roots of the distance polynomial Ch(D,G). Using the Le Verrier-Fadeev-Frame algorithm, Balasubramanian computed the distance polynomials for a large collection of fullerenes. The distance polynomial Ch(D,46) and spectrum Sp(D,46) of the Hiickel graph of the fullerene C20 (/ ), graph 46, are presented in Tables 10 and 11, respectively. 

The distance polynomials and spectra of 49 computed with the X, Y, and Z weighting schemes are ... 

The Ch(RD,C) polynomial is a polynomial with real number coefficients. Because the distance polynomial coefficients have large positive or negative values, the structural descriptors based on the distance polynomial will have too large values to be useful descriptors in QSPR or QSAR studies. The coefficients of the reciprocal distance polynomial have real values, small enough to be able to generate structural descriptors which can be used in structure-property studies. The spectrum Sp(RD.G) represents the set of eigenvalues of the RD matrix, or the roots of the Ch(RD,G) polynomial. 

Those based on the topological distance matrix, including the Wiener index, the polarity number, the distance sum, the Altenburg polynomial, the mean square distance, the Hosoya index, and the distance polynomial ... 

An alternative way to eliminate discontinuities in the energy and force equations is to use a switching function. A switching function is a polynomial ftmction of the distance by which the potential energy function is multiplied. Thus the switched potential o (r) is related to the true potential t> r) by v r) = v(r)S(r). Some switching functions are applied to the entire range of the potential up to the cutoff point. One such function is ... 

As a point P(x,y) on a curve moves away from the region of the origin (Figure 1-36), the distance between P and some fixed line may tend to zero. If so, the line is called an asymptote of the curve. If N(x) and D(x) are polynomials with no common factor, and... 

This relation for the thickness of the boundary layer has been obtained on the assumption that the velocity profile can be described by a polynomial of the form of equation 11.10 and that the main stream velocity is reached at a distance 8 from the surface, whereas, in fact, the stream velocity is approached asymptotically. Although equation 11.11 gives the velocity ux accurately as a function of v, it does not provide a means of calculating accurately the distance from the surface at which ux has a particular value when ux is near us, because 3ux/dy is then small. The thickness of the boundary layer as calculated is therefore a function of the particular approximate relation which is taken to represent the velocity profile. This difficulty cat be overcome by introducing a new concept, the displacement thickness 8. ... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

Figure 8-3. fjj (r/j) as cubic splines. The distance, rjj, between atoms i and j is divided into the mesh as shown. In each mesh, the pair-force fjj is modeled as a cubic polynomial... 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

Thus the distance to the end of the n-th rod is obtained by n-fold convolution of the rod length distribution. A typical series of such lattice constant distributions is demonstrated in Fig. 8.43. Its sum is named convolution polynomial. 

Prove that the Legendre polynomial / ( ) has the smallest distance in the mean from zero of all polynomials of degree n with lending coefficient 2"( i)ni o ... 

Thangavel, P and P. Venuvanalingam, Algorithm for the Computation of Molecular Distance Matrix and Distance Polynomial of Chemical Graphs on Parallel Computers. J. Chem. Inf. Comput. Sci., 1993 33, 412—414. 

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