Matrix bond 16 molecular graph

Central the molecular graph is completely coded (each atom and bond is represented) matrix algebra can be used the niimber of entries in the matrix grows with the square of the number of atoms in ) no stereochemistry included... 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

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. 

Fig. 3.2 Schematic of the CATS descriptor calculation, (a) The hydrogen-depleted two-dimensional molecular graph provides the input, (b) The graph is simplified for the distance matrix computation different bond orders are not considered (unweighted graph) and all element types are disregarded. The algorithm starts at an arbitrary chosen atom and visits all nodes of the graph in a breadth-first manner, thereby building up the distance matrix. The numbers at the vertices are used to reference individual atoms in the distance matrix.

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. 

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. 

The total adjacency index Ay is the sum of all the entries of the adjacency matrix of a molecular graph, and is twice the - bond number B [Harary, 1969a] ... 

The -> adjacency matrix A of a molecular graph G is an example of binary sparse matrix, only the off-diagonal entries i-j, where v, and Vy are adjacent vertices, i.e. vertices connected by a bond, being equal to one. Using the adjacency matrix as multiplier in the Hadamard product it follows ... 

This is the simplest graph invariant obtained from the -> adjacency matrix A, defined as the number of bonds in the -> molecular graph (7 where multiple bonds are considered as single edges. Bond number is calculated as half the - total adjacency index Ay ... 

They are topological rectangular matrices derived from a -> molecular graph Q where each column represents a graph - circuit. Two main cycle matrices are defined the vertex cycle matrix Cv whose rows are the A vertices (i.e. the atoms) and the edge cycle matrix Ce whose rows are the B edges (i.e. the bonds) of the graph [Bonchev, 1983]. 

A rectangular cycle matrix whose rows are the edges (bonds) and columns the circuits of the graph, i.e. having a dimension B x C+, where C is the cyclicity. Derived from H-depleted molecular graph, its elements are Cy = 1 if the i th edge belongs to the j th circuit, otherwise c,y = 0. 

Derived from the -> H-depleted molecular graph, the incidence matrix [Bonchev and Trinajstic, 1977] is a rectangular matrix representation of a graph whose rows are the vertices (atoms, A) and columns are the edges (bonds, B), i.e. having a dimension AxB. Their elements are i,y = 1 if the edge ey is incident to the vertex V , otherwise... 

The path-distance map matrix, denoted as PD, resembling the bond length-weighted distance matrix of a molecular graph, is defined as [Bajzer, Randic et al, 2003]... 

Derived from the —> molecular graph Q, the edge adjacency matrix, denoted by E, or more formally as A, also called bond matrix, encodes information about the coimectivity between... 

It is a square symmetric matrix of dimension B x B, where B is the number of bonds, and is usually derived from a H-depleted molecular graph [Bonchev, 1983]. It is to be noted that the edge adjacency matrix of a graph Q is equal to the adjacency matrix of the line graph of Q [Gutman and Estrada, 1996]. 

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