Extended Adjacency Matrices

The extended vertex-adjacency matrix, denoted by E A, is a square symmetric VxV matrix defined as (Yang et al 1994) 

From Equation (2.28) is also evident that the extended edge-adjacency matrix is also a sort of the edge-weighted adjacency matrix. 

The extended edge-adjacency matrix of G, is equal to the extended vertex-adjacency matrix of L(Gi). 

Topological indices based on the extended edge-adjacency matrix have not yet been explored in QSPR or QSAR modeling. 

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They are two molecular descriptors calculated from -+ extended adjacency matrices EA [Yang et al, 1994], The first is the sum of the absolute eigenvalues of the EA matrix, called the EAI index ... 

The extended adjacency matrices EA are - weighted adjacency matrices Ax.A whose elements are defined as a function of local vertex invariants of the -> adjacency... 

H -state index = hydrogen electrotopological state index electrotopological state indices > heteroatom-corrected extended adjacency matrix —> extended adjacency matrices... 

Related to the % matrix are also the —> extended adjacency matrices based on a sort of average vertex degree and the edge-XI matrix, which is derived from —> distance-valency matrices Dval where the simple vertex degree of the % matrix is replaced with the valency of a vertex [Ivanciuc, 1999c] ... 

Other important weighted vertex adjacency matrices are the extended adjacency matrices, the edge-Wiener matrix, —> edge-Cluj matrices, —> edge-Szeged matrices, and the random walk Markov matrix. 

Extended Adjacency Matrix-Kow Relationships Yang et al. [39] derived two descriptors, EAS and EAmax, from the extended adjacency (EA) matrix and demonstrated their correlation with log,0 ow for barbiturate acid derivatives with structures I and II ... 

EA indices Extended Adjacency matrix indices -> eigenvalue-based descriptors... 

EAI index -> eigenvalue-based descriptors (O extended adjacency matrix indices) eccentric distance matrix eccentric connectivity index ( )... 

A further extension of the extended adjacency matrix, in terms of conventional bond order n, can also be made to consider bond multiplicity the entries of the heteroatom/multiplidty-corrected extended adjacency matrix EA are the following ... 

From the extended adjacency matrix defined above, two - eigenvalue-based descriptors, called -> extended adjacency matrix indices, were proposed as molecular descriptors. 

A complex ID number calculated by the matrix powers of an extended adjacency matrix, derived by weighting the vertices of the graph [Hu and Xu, 1996]. 

An extended adjacency matrix EA is defined here by the following ... 

EA indices = Extended Adjacency matrix indices spectral indices > EAmax index spectral indices (0 extended adjacency matrix indices)... 

The extended vertex adjacency matrix (or simply extended adjacency matrix) is an adjacency matrix EA whose entries are defined as... 

A correction factor can be introduced to account for heteroatoms, such as the —> atomic electronegativity xy, then, the entries of the heteroatom-corrected extended adjacency matrix E A are the following ... 

Recently Hu and Xu proposed a highly discriminating molecular topological index based on the extended adjacency matrix. The extended adjacency matrix is computed by... 

Thus, any spectral moment and hence the activities/properties of chemical compounds can be represented by contributions of corresponding fragments. This approach was further extended to molecular graphs containing heteroatoms by weighting the diagonal elements of the bond adjacency matrix. 

It must be pointed out that the extended connectivity EC of Morgan cannot be considered a new local invariant as it coincides with the - atomic walk count awc calculated as row sum of the /cth power of the adjacency matrix A [Razinger, 1982 Rucker and Rucker, 1993 Figueras, 1993]. 

The procedure for counting S3 and D3 3 is based on the extended vertex-adjacency matrix and the adjacency bonding array. The extended vertex-adjacency matrix of a Kekule structure contains elements 1 or 2 depending on the single or double bond between carbon atoms in the Kekule structure and it is called extended because of the added column and row in which elements are set to 1 or 0 depending whether the carbon atom in the Kekule structure is connected to the hydrogen bond or not. For example, the extended vertex-adjacency matrix for the Kekule structure of benzene (see Figure 15) is ... 

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