Hosoya matrix

Randic, M. (1994b). Hosoya Matrix - A Source of New Molecular Descriptors. CroatChem. Acta, 67,415-429. 

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

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). 

M. Randid, Hosoya matrix—A source of new molecular descriptors, Croat. Chem. Acta 67 (1994a) 415 29. 

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 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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