Distance polynomial

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

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. 

Diudea, M.V, Ivanciuc, O., Nikolic, S. and Trinajstic, N. (1997c). Matrices of Reciprocal Distance, Polynomials and Derived Numbers. MATCH (Comm.Math.Comp.Chem.), 35, 309-318. Diudea, M.V, Katona, G. and Parv, B. (1997d). Delta Number, of Dendrimers. Croat. Chem.Acta, 70, 509-517. 

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] ... 

Hosoya, H Murakami, M. and Gotoh, M. (1973) Distance polynomial and characterization of a graph. Natl. Sci. Kept. Ochanomizu Univ., 24, 27-34. 

Thakur, M., Thakur, A. and Khadikar, P.V. (2004b) QSAR studies on psychotomimetic phenylalky-lamines. Bioorg. Med. Chem., 12, 825-831. Thangavel, P. and Venuvanalingam, P. (1993) Algorithms for the computation of molecular distance matrix and distance polynomial of chemical graphs on parallel computers. /. Chem. Inf. Comput. Sci., 33, 412-414. 

These kind of graphs are called twin graphs (Hosoya et al., 1994, 2001), because they possess, besides identical distance-spectra and consequently identical distance-polynomials, identical characteristic polynomials and their spectra (3.8801, 1.3557, 0.7732, 0.4773, -0.7376, -1.2464, -2.0953, -2.4069), identical matching polynomials and their spectra, and many identical graph-theoretical invariants. 

H. Hosoya, Distance polynomial and the related counting polynomials, Croat. Chem. Acta 86 (2013)443M51. 

M. Diudea, T. Ivanciuc, S. Nikolic, and N. Trinajstic, Matrices of reciprocal distance, polynomials and derived numbers, MATCH Commun. Math. Comput. Chem. 35 (1997) 41-64. 

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 ... 

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