The Hosoya Matrix

The Hosoya matrix was introduced by Randid (1994a). Denoted by Z, the Hosoya matrix is derived here in a manner similar to the edge-Wiener matrix (see Section 5.3). It is a sparse symmetric square VxVmatrix whose elements for a tree are defined as 

The Hosoya matrix may be made dense if the elements [Z],y are computed not only for deleted edges, but also for deleted edges along any path in a tree (Randic, 1994a). Then, the dense Hosoya matrix Z is defined as 

The Hosoya matrices are used to produce a variety of molecular descriptors, especially since the Z-index and the Hosoya matrix have been extended to polycyclic systems and edge-weighted graphs (e.g., Nikolid et al., 1992 Plavsid et al., 1997 Espeso et al., 2000 Milicevic et al., 2003). Randid (1994a) tested successfully the Z-indices in the structure-boiling point modeling of octanes. Similarly, Hosoya (2002) used his index for predicting the octane numbers of heptanes and octanes. Mathematical properties of the Hosoya Z-indices have also been studied (e.g., Vukicevid and Trinajstid, 2005). 

Hosoya elaborated the potentiality of his descriptor in his talk at the International Symposium on Applications of Mathanatical Concepts in Chemistry (Dubrovnik, Croatia, September 2-5, 1985) and in his article in the book entitled Mathematics and Computational Concepts in Chemistry (Hosoya, 1986). Hosoya also produced in 2012 a monograph on his index (Hosoya, 2012), published in Japanese. Perhaps it will be translated into English. 

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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 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. 

A square symmetric matrix of dimension AxA, A being the munber of vertices in the -> H-depleted molecular graph Q. The original Hosoya Z matrix is defined only for acyclic graphs its entry Zy is equal to the Hosoya Z index of the subgraph G... 

A general definition of the Hosoya Z matrix (generalized Hosoya Z matrix) able to represent both acyclic and cyclic graphs is the following [Plavsic et al, 1997] ... 

The Z7Z index is among the graphical bond order descriptors and can be obtained from the Hosoya Z matrix only by considering the entries relative to adjacent vertices (i.e. bonds) ... 

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] ... 

By analogy with the Hosoya Z index that, for acyclic graphs, can be calculated as the sum of the absolute values of the coefficients of the characteristic polynomial of the adjacency matrix, the stability index (or modified Z index) is a molecular descriptor calculated for any graph as the sum of the absolute values of the coefficients C2i appearing alternatively in the characteristic polynomial of the adjacency matrix [Hosoya, Hosoi et al., 1975] ... 

Characteristic polynomial, spectrum, spectral moments, eigenvectors, and Hosoya-type indices were also computed on square molecular matrices encoding information about spatial interatomic distances such as the geometry matrix G and the reciprocal geometry matrix [Ivanciuc and Balaban, 1999c]. 

For acyclic graphs the Z-counting polynomial coefficients a((5, k) coincide with the absolute values of the coefficients of the characteristic polynomial of the adjacency matrix (i.e., graph characteristic polynomial) [Nikolic, PlavSic et al., 1992]. Therefore, for any graph, the Hosoya Z index can also be calculated from the matching polynomial coefficients m2k as... 

The diagonal entries are zero by definition [Randic, 1994b], If more than one subgraph is obtained by the erasure procedure, the matrix element is calculated by summing up all of the Hosoya Z indices of the subgraphs. 

Other graph invariants derived from the Hosoya Z matrix are the eigenvalues and the coefficients of the —> characteristic polynomial. Moreover, sequences of weighted paths and the weighted path counts were defined using as the path weights the magnitude of the Z... 

B and C are the number of vertices and the cyclomatic number, respectively, and ay the elements of the adjacency matrix, taking values equal to 1 for pairs of adjacent vertices, and zero otherwise. Other molecular descriptors were calculated by applying several different matrix operators [Ivanciuc, 2000h], such as spectral indices, Hosoya-like indices, and Balaban-like information indices. 

Usually, the Wiener index is defined and calculated as the sum of all topological distances in the H-depleted molecular graph [Hosoya, 1971]. It is obtained from the distance matrix D... 

Espeso, V.G., Molins Vara, J.J., Roy Lazaro, B., Riera Parcerisas, F. and Plavsic, D. (2000) On the Hosoya hyperindex and the molecular indices based on a new decomposition of the Hosoya Z matrix. Croat. Chem. Acta, 73, 1017-1026. 

Local vertex invariant based on the adjacency matrix, atomic numbers, and vertex degrees Mean distance topological index for any graph Mean distance topological index for acyclic graphs Hosoya index... 

Hosoya (2013) and Hosoya et al. (1994, 2001) observed that two nonisomorphic graphs may possess identical distance-spectra. We already mentioned isospectral graphs when presenting the Hiickel matrix (see Section 2.19). A pair of the two polyhedral graphs on eight vertices that possess the same distance-spectra are shown in Figure 4.1. 

The dense Hosoya matrix for T2 (see Figure 2.19) is exanplified below ... 

The third numerical matrix is named the sparse vertex-Hosoya matrix and denoted by Z. An example of this matrix obtained from the sparse vertex-graphical matrix of Tj is given below. Only the upper matrix-triangle is shown. 

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