Matrices, derived from adjacency matrix

A bond distance-weighted edge adjacency matrix is derived from the edge adjacency matrix using - bond distances calculated by computational chemistry methods as diagonal entries on selected molecular geometries [Estrada, 1997] ... 

The additive adjacency matrix is derived from the adjacency matrix substituting row elements equal to 1, corresponding to pairs of adjacent vertices, with the vertex degrees of the coimected vertices as... 

Each predefined unit is related to its adjacency matrix representation. Finally, the derived units are assembled thus, the structural information is derived from the compound name. 

The distance matrix D(G) of a graph G is another important graph-invariant. Its entries dy, called distances, are equal to the number of edges connecting the vertices i and j on the shortest path between them. Thus, all dy are integers, including du = 1 for nearest neighbours, and, by definition, d = 0. The distance matrix can be derived readily from the adjacency matrix ... 

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

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 ... 

First described in 1979 as secreted phosphoprotein 1 (Senger et al., 1979), OPN has a shorter history but shares with MIF some of its mystique. The name osteopontin derives from the fact that the protein was found to be present in the bone, and as it is localized in the osteoid matrix, it is believed that it can form a bridge between bone and the adjacent cellular tissues (Oldberg et ah, 1986). 

Derived from the -> molecular graph <5, the adjacency matrix A represents the whole set of connections between adjacent pairs of atoms [Trinajstic, 1992]. The entries Oy of the matrix equal one if vertices v, and Vy are adjacent (i.e. the atoms / and j are bonded) and zero otherwise. The adjacency matrix is symmetric with dimension A x A, where A is the number of atoms and it is usually derived from an -> H-depleted molecular graph. 

Other topological matrices are derived from the adjacency matrix, such as -> atom connectivity matrices, -> Laplacian matrix and the powers of the adjacency matrix used to obtain walk counts and the corresponding - molecular descriptors. 

From the edge adjacency matrix, a graph-theoretical invariant analogous to the -> Randic connectivity index was derived by Estrada [Estrada, 1995a] it is called edge connectivity index, denoted by e, and defined as ... 

More interesting is the -> edge-type Schultz index derived from both the edge distance matrix and the edge adjacency matrix. 

Spectral moments of iterated line graph sequence were derived from the - adjacency matrix A of the line graph L of any order n as ... 

Among the LOVIs obtained by this general approach, the most known are AZV descriptors derived from the adjacency matrix A whose diagonal elements are substituted by the atomic numbers Z, and the A-dimensional vector r contanining the vertex degrees 8j. 

For simple structures, the path counts can be derived directly from the molecular graphs otherwise specific algorithms are needed. For example, Randic s algorithm results in path counts for nonequivalent vertices from the -> adjacency matrix [Randic etal, 1979]. 

Index, derived from the -> adjacency matrix A, the - distance matrix D and the A-dimensional column vector v constituted by the vertex degree b of the A atoms in the -> H-depleted molecular graph [Schultz, 1989]. The Schultz index is defined as ... 

An edge-type Schultz index has been derived from the - edge adjacency matrix E and the - edge distance matrix [Estrada and Gutman, 1996 Estrada and Rodriguez, 1997] ... 

Derived from the adjacency matrix A, this is defined as ... 

This is derived from the partition of the adjacency matrix A elements according to their magnitude and is a trivial quantity, as zero entries do not contribute to the - total adjacency index Av, i.e. the sum of all vertex degrees, corresponding to twice the bond number 5 ... 

Derived from the adjacency matrix A and based on the partition of vertices according to - vertex degree equality, this is defined as ... 

Bone tissue also contains osteoclasts, which are multinucleated cells that are derived from the hematopoietic (granulocyte-monocyte) cell line located in bone marrow. Osteoclasts are primarily responsible for bone resorption and they secrete enzymes and hydrochloric acid that break down collagen matrix and help dissolve the bone. The area where osteoclast cell membrane lies adjacent to bony tissue is known as a Howship s lacunae. The osteoclast cell membrane that lies in close proximity to bone can contain numerous villous extensions and form a ruffled border. These areas are also known as resorption pits. The plasmalemma border of the osteoclast cell in this region forms a specialized seal with the underlying bone to prevent the release of enzymes and hydrochloric acid. This process also results in the release of growth factors previously deposited in bone by osteoblasts, which are responsible for maintaining the process of regenerating new bone. 

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