Node of graph

If the nodes of a graph are marked (e.g., witli digits), the graph is termed labeled. In the example, node I is adfacent to node 2 bnt not to node 3. 

WriE69 Wright, E. M. The number of graphs on many unlabelled nodes. Math. Annal. 183 (1969) 250-253. 

One may question whether the application of graph theory is really necessary as no doubt a separation such as that described above can be investigated easily without it. However, when more ions are to be separated, the number of nodes grows very rapidly. 

DEFINITION Let G be a directed labelled graph and G a subgraph of G. We call G a single entry subgraph of G if there is a node n in G, called the entry node of G such that... 

Here A is a single entry subgraph with entry node and a self-loop back to n. We now add a single entry subgraph A graph isomorphic to A and connect A to n, , the entry of A. There are two possibilities. If node m in A was connected to, then the corresponding node m in A can be connected either back to the entry node of A or A or to the entry node n of A. ... 

In the first step a Shortest Path-Algorithm (Dijkstra or A ) calculates one or more shortest paths between two given nodes of the plant graph [5],... 

Let the nodes of the graph represent process variables and the edges the relationships (balance equations) between them. There is a directed edge from node a to node i, if a belongs to the interval of i, i.e., if we need a to evaluate i. 

Fig. 4. A schematic illustration of graph comparisons. The purpose of comparing different types of graphs is to identify correlated clusters of nodes.

The Morgan algorithm 5> is a device used by the Chemical Abstracts Service for assigning indices to the nodes of constitutional formulas whose H-atoms have been omitted, i. e. their reduced graphs. If the indices of all atoms are needed, e.g. for the representation of stereochemistry, an additional procedure is needed for establishing those for the H-atoms. Constitutional symmetry is not indicated directly by Morgan indices. 

The reacting system can be represented by graphs (trees) In which the nodes represent monomer units. In the theory of branching processes this collection of graphs (Figure 2) - a molecular forest -Is transformed Into another forest - the forest of rooted trees. 

The Method of Volkenstein and Goldstein. Volkenstein and Goldstein have applied the theory of graphs to the derivation of rate equations. Their approach has three main features the use of an auxiliary node, the compression of a path into a point, and the addition of parallel branches. These can be best explained by an example (Scheme 3). 

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