Graph Operations

Assumes graph operating conditions (not provided) ate the same as associated text of Ref. 15. 

An adjacency matrix representation of a graph has several nice properties. Many natural graph operations correspond to standard matrix operations (see (5) for some examples). The bits of M can be packed in groups into computer words, so that... 

Fig. 16. Example of tree-like representation for RNA secondary structure. Each hairpin structure is shown next to its equivalent tree. With such representations, a graph theoretic measure can measure the distance between these trees and help generate fitness values for a fitness landscape. For example, the distance between two structures may be defined as the minimal number of elementary graph operations (insert a point, switch an edge, etc.) needed to convert one tree into the other. Note that there are many variants of tree representations for RNA secondary structures and many definitions of graph distance. In low-resolution tree representations, several secondary structures can map to the same graph.

Polanski, J. and Bonchev, D. (1986). The Wiener Number of Graphs. I. General Theory and Changes Due to Graph Operations, MATCH (Comm.Math.Comp.Chem.),21,133-186. Polanski, J. and Bonchev, D. (1987). The Minimum Distance Number of Trees. MATCH (Comm. Math.Comp.Chem.), 21, 314-344. 

Ivanciuc, O. and Balaban, A.T. (1999b) Design of topological indices. Part 21. Molecular graph operators for the computation of geometric structural descriptors. Rev. Roum. Chim., 44, 539-547. 

Other basic graph operations are functions to compute the representation of cofactors from the graph representation of F by replacing... 

Virtually any graph operation is implemented as a recursive function that consists of a termination test and a recursive part (the Apply step). Whenever only a single graph type is involved, algorithms from the original work can be used. Otherwise, arguments of an operation are expanded recursively until a return value is obtained. If the graph type of the result is different from the... 

Figure 2 outlines the Apply step based on theorem 1. The recursive function computes the graph representation W of function r with graph type r. are graph representations of type and rp for functions / and (/, respectively. denotes an arbitrary binary graph operation. The algorithm assumes, that... 

While this algorithm is sufiicient to compute any binary graph operation, it can be improved depending on operation . Simplified versions exist for addition and multiplication in the domain of function r. In both cases, computation of cofactors (lines 8,9 in figure 2) as well as computation of the successor function succ (line 10) can be avoided. 

Graph operation Path complement Path addition... 

Graph operation Path addition Path nesting... 

Graph operation Path addition Operator embedding... 

Observe that the subgraphs as well as G are determined by the original graph G[N,J] with the partition J = J° u and by the prescribed algebraic graph operations the drawings are only illustrations for the reader. 

The problems of solvability arise when certain mass flowrates are given a priori see Section 3.2. The graph structure allows one to analyze the problems completely by graph operations. The arc set J is partitioned... 

DIS), an extensible object-oriented database model, which has also been used for VLSI/CAD application environments (see and ). The 3DIS provides an approach in which data and the descriptive information about the data are handled in a uniform framework. It is especially suited to represent information objects of various levels of abstraction and modality, and supports various views of the same data which is essential for this type of engineering database. In addition to that, the EDEN system supports a number of novel modeling constructs, one of which makes it possible to interpret parts of the data as graphs and to express retrievals, constraints and updates in terms of graph operations. 

The consistency procedure mentioned above can be easily defined in terms of the general graph operations. To find a transformation between two not directly connected frames a path is found first, and than a generic distance is calculated, where h returns the matrix and g performs matrix multiplication. The consistency check consists of comparing the result with the matrix that has to be added to the database. 

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