Incidence matrix

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... 

Note that the incidence matrix and the transpose of the circuit matrix (keeping the edges in the same order) are orthogonal modulo two i.e., AB = 0.8... 

There is an appropriate way of generating the cut-set matrix from the incidence matrix that requires considerable detail and involves the additional idea of a tree and of fundamental circuits and cut-sets. 

It is clear that an incidence matrix is associated with each graph. 

In the foregoing discussion the properties of the incidence matrix and the cycle matrix were illustrated in terms of a cyclic digraph, but the results on the ranks of these matrices actually hold true for any connected digraph with N vertices. For an undirected graph, M and C contain only 0 and 1 (sometimes referred to as binary matrices), mathematical relations of identical form are obtained except that modulo 2 arithmetic2 is used instead of ordinary arithmetic. The ranks of M and defined in terms of modulo 2 arithmetic are JV — 1 and C, as before, and Eqs. (10) and (11) are modified to read... 

Equation (6a) shows the incidence matrix M corresponding to the graph in Fig. 2a, and Eqs. (8a) and (8c) show two cycle matrices corresponding to the same graph. 

Just as the incidence matrix M is related to material balances around each of (N - 1) vertices, the cut-set matrix K is clearly related to material balances around the (N - 1) vertex subsets VK (or VB, since material conservation holds over the whole network). In this case the (N — 1) vertex subsets happen to be a, e, c, (d, f, and f. ... 

For computational purposes it is useful to distinguish the directed edges associated with the external inputs and outputs, which are usually specified, from those associated with the flows internal to the network. Denoting the net output from vertex i by and including henceforth only the internal edges in the incidence matrix M, we may restate Eq. (7) as... 

M Incidence matrix. M refers to a digraph (6b), and fil refers to an undirected graph (6a)... 

The incidence matrix of such a graph specifies the connections of vertices and edges. For species J (figure 2) one has ... 

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

Further, if A is still the incidence matrix of T and the adjacency matrix of A, then f(E,V) is a function which transforms a given caterpillar into the associated line graph [5], i e ... 

Fig 10 Walk codes and equivalent contours possessing the same incidence matrix A of the indicated comb caterpillar ... 

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