The Hosoya Index

Alternatively, for acyclic graphs Z can be defined as the sum of the absolute values of coefficients in the characteristic polynomial PH(G, x)  

N is the number of atoms, A and E are the adjacency and unit matrix, respectively, while s is the largest number of edges disconnected to each other in the acyclic graph. Thus, for Gx one obtains  

The characteristic polynomial may be obtained readily from the adjacency matrix A by placing x s on the main diagonal and expanding the determinant (39). 

We classified Z in the group of indices associated with the distance matrix (and not to the adjacency matrix, as Trinajstic did 21-22 ) due to the procedure for counting p(G, k). 

The Hosoya index was applied 2,60 63) to correlations with boiling points, entropies, calculated bond orders, as well as for coding of chemical structures. 

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In addition to these, only a limited number of other topological indices of benzenoid molecules have been studied. With a few not too important exceptions, generally valid mathematical results were obtained only for one of them — namely for the Wiener index. Therefore the remaining part of this section is devoted to the Wiener index of benzenoid systems. (Further graph invariants worth mentioning in connection with benzenoids, especially unbranched catacondensed systems, are the Hosoya index [119-121], the Merrifield — Simmons index [122, 123], the modified Hosoya index [38] and the polynomials associated with them.)... 

Gutman, I. (1988). On the Hosoya Index of Very Large Molecules. MATCH (Comm.Math.-Comp.Chem.), 23,95-104. 

Gutman, I. (1988b) On the Hosoya index of very large molecules. MATCH Commun. Math. Comput. Chem., 23, 95-104. 

Gutman, 1., Kolakovic, N., Graovac, A. and Babic, D. (1989b) A method for calculation of the Hosoya index of polymers, in MATH/CHEM/COMP 1988 (ed. A. Graovac), Elsevier, Amsterdam,... 

Other well-established indices include the Zagreb index (the sum of the squared vertex valencies, where valency means number of connections to heavy atoms, regardless of bond order) and the Hosoya index [51]. New indices are published regularly to add to the many hundreds that already exist. Many are highly correlated with others procedures to reduce redundancy are discussed in Sec. 3. 

Garg and Achenie also demonstrated a reasonable approach to the solution of the I-QSAR problem in 2001. " ° Taking a target scaffold of an antifolate molecule for dihydrofolate reductase inhibition, these authors generated a QSAR for both activity and selectivity. They solved the I-QSAR problem to maximize selectivity through changing substitutents on the scaffold, subject to a constraint of a threshold activity. Finally, a work by Skvortsova, et al. " from 2003 demonstrated that the I-QSAR problem could be solved for the Hosoya index plus constraints on the number of carbon atoms for a system of 78 hydrocarbons. 

Des., 2, 70 (2003). Molecular Design of Chemical Compounds with Prescribed Properties from QSAR Models Containing the Hosoya Index. 

The first numerical matrix, using the Hosoya index, is called the edge-Hosoya matrix and is denoted by Z. This matrix was already discussed in the Section 5.12, where it was called simply the Hosoya matrix. If we sum the elements in one triangle of the Z matrix as originally suggested by Hosoya (1971) when he defined the Wiener index from the distance matrix, the double invariant so obtained is called the edge-Hosoya-Wiener index. 

The characteristic polynomial is related to another important polynomial of interest in physics and chemistry, the matching polynomial, the coefficients of which count independent sets of edges in a graph [1]. When added, these counts give the Hosoya index Z [2], one of the early molecular descriptors for structure-property regressions. 

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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