Matrix, cycle

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... 

The edges in a spanning tree are called tree branches or branches. All other edges of G are called chords. Thus, with reference to Tu the chords are 6, 7, 8. Because there is one and only one path between any two vertices of Tj, the addition of any chord to Tx will create exactly one cycle. Such a cycle is called a fundamental cycle. It follows that there are as many fundamental cycles as there are chords (P — N + 1 = C). Thus for the graph in Fig. 1 the fundamental cycles are 3, 4, 6, 2, 4, 7, and 2, 4, 5, 8). Notice that the fundamental cycles are defined only with respect to a given spanning tree. If more than one chord is added to Tx at the same time, cycles which are not fundamental cycles will also be created. For instance, simultaneous addition of chords 7 and 8, will create not only the last two fundamental cycles but also 5, 7, 8 which is not a fundamental cycle. Since each chord occurs only once in a set of fundamental cycles, it should be evident that the rows of a cycle matrix corresponding to the fundamental cycles will be linearly independent and the rank of the cycle matrix will be (P — N + 1). Such a matrix will be referred to as a fundamental cycle matrix. 

Since the permutation of rows and columns is immaterial, we can always arrange the columns so that the first (N — 1) correspond to the tree branches and the last C correspond to the chords. Hence a fundamental cycle matrix f can always be written as... 

The cycle matrix of Table XXII is a tabulation of mechanism (43) with p = 0, a = 0, and t = 0, and the row vector (51) consists of the coefficients in (43) with = 0, x = 0, and ij/ = 0. Any three independent cycles could have been chosen to generate Table XXII and any mechanism for the overall reaction could have been chosen to establish the row vector (45). The choices we made are arbitrary and depend on the diagonalization procedure used to find the matrix of Table XXI, which is far from unique. The important point is that the list of direct mechanisms we are looking for is unique and independent of how the above choices are made. 

The matrix K(a) is the start-of-cycle matrix K at t = 0, appropriately modified by the a s. Equation (40) represents the complete reformer model kinetic expressions. 

Partitioning and the Cycle Matrix. Sequential modular systems require an order of calculation (precedence order) be given to the modules. There are generally four steps taken to determine this ordering. 

One of the basic tools of analysis for tearing is the cycle matrix. This consists of a matrix of streams (the row) and the loop in which they are contained in columns. For example, consider Figure 3 (the Cavett problem). The cycle matrix is found by placing a 1 in the loop column if a stream appears in a loop or 0 if it does not. This is shown in the cycle matrix of Figure 4. The total number of loops in which a stream is included is calculated and placed in the loop total column. We shall use this number later. Note that the loop must pass through a module (node) only once. 

Figure 5 gives a system where it is not possible to tear each loop only once and Figure 6 gives its cycle matrix. 

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]. 

Based on total and mean information content, several - topological information indices can be calculated both on the vertex cycle matrix (- information indices on the vertex cycle matrix) and the edge cycle matrix (- information indices on the edge cycle matrix). 

A rectangular cycle matrix whose rows are the vertices (atoms) and columns the circuits of the graph, i.e. having a dimension AxC, where C is the cyclicity, i.e. the number of circuits. Derived from the -> H-depleted molecular graph, its elements are Cij = 1 if the vertex v, belongs to the yth circuit, otherwise Cy = 0. 

Collected below are the information indices calculated on the most important topological matrices such as - adjacency matrix A, distance matrix D, - edge distance matrix "D, - edge adjacency matrix E, - vertex cycle matrix Cy, and -+ edge cycle matrix Ce [Bonchev et al, 1981b Bonchev, 1983]. 

Information indices on the vertex cycle matrix Cyare listed below. 

This is derived from the vertex cycle matrix Cy and is based on the partition of matrix elements according to their equalities ... 

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