Bond incidence matrix

Bond incidence matrix for benzene using traditional single and double bonds... 

The incidence matrix is an n A m matrix where the nodes (atoms) define the columns (n) and the edges (bonds) correspond to the rows [m). An entry obtains the value of 1 if the corresponding edge ends in this particular node (Figure 2-16),... 

Figure 2-16. a) The redundant incidence matrix of ethanal can be compressed by b) omitting the zero values and c) omitting the hydrogen atoms, in the non-square matrix, the atoms are listed in columns and the bonds in rows. 

Incidence matrix describes connections and bonds contains only 0 and L(bits) no bond types and bond orders no number of electrons... 

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 third weight is related to vertex valence and takes into account the number of bonds incident to a vertex and its pairs of unshared electrons. Each present electron pair counts one, each missing electron pair contributes -1 to the total valence of heteroatoms, and in the case of free radicals the unique electron present in the outer valence shell contributes half a bond to the total valence. The valence weights are collected into a diagonal matrix W used to give a valence weighted distance matrix D ... 

Information about atoms and bonds belonging to cycles is usually encoded by the vertex-cycle incidence matrix and the edge-cycle incidence matrix. 

As each conserved state determines a domain, additional connection constraints can be found for various port types. For instance, a bond connected to one side of a 0-junction may be connected to a C-type storage port or a source port, as these ports do not violate the balance equation. However, in principle, one should be more careful when connecting an I-type, R-type, TF-type, or GY-typeport, because these ports cannot absorb the conserved state related to the flow. However, all domains with relative equilibrium-determining variables have a non-displayed balance for the reference node (this balance equation is dependent on the balance equations for the rest of the network and corresponds to the row that is omitted in an incidence matrix to turn it into a reduced incidence matfix of an electrical circuit, for example). This additional balance compensates for this flow, such that it is still possible to connect these ports without violating the balance equation. Note that the I-type port in principle is a connection to a GY-type port that connects to the storage in another domain. Some domains have absolute equilibrium-determining variables, like temperature and pressure, but since in most cases it is not practical to choose the absolute zero point as a reference, usually another reference state is chosen, such that these variables are treated as differences with respect to an arbitrary reference and an additional balance too. 

The matrix of a structure with n atoms consists of an array of n / u entries. A molecule with its different atoms and bond types can be represented in matrix form in different ways depending on wbat kind of entries are chosen for the atoms and bonds. Thus, a variety of matrices has been proposed adjacency, distance, incidence, bond, and bond-electron matrices. 

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