Vertex set

I Vy 1= h V) depends on the partition. Let V) be the number of nodes in the set Vi- Then the complexity X G) of the graph G is given by minimizing Shannon s information over all possible partitions of this vertex set ... 

A nonempty set V of graph vertexes forms a sink, if there are no oriented edges from At eV to any Ay V. Eor example, in the reaction graph A] <— A2 A3 the one-vertex sets A] and A3 are sinks. A sink is minimal if it does not contain a strictly smaller sink. In the previous example, A] and A3 are minimal sinks. Minimal sinks are also called ergodic components. 

To construct an approximation to the relaxation process in the reaction network iV, we also need to restore cycles, but for this purpose we should start from the whole glued network V on si (not only from fixed points as we did for the steady-state approximation). On a step back, from the set si to si and so on some of glued cycles should be restored and cut. On each step we build an acyclic reaction network, the final network is defined on the initial vertex set and approximates relaxation of if. 

Proof Let us prove (i) we can assume that G is distinct from ( 4,7, 3)spec. The first step consists of associating to G another 4-valent plane skel(G), whose vertex-set consists of all 4-gons. Every 7-gonal face is adjacent to two 4-gons hence, it defines an edge of skel(G) and skel(G) is 4-regular. See below some representations of the local structure of G (in straight lines) and skel(G) (in dashed lines) ... 

But, if this happens, then we have two cycles that are not homologous to 0. Then we consider the graph Conn(G), whose vertex-set consists of connected... 

The process continues until either 1 vertices are matched or there is found to be no way of matching the vertex sets of the two graphs. 

There is no problem in identifying the vertex set in a molecular graph that represents the constitutional formula of a molecule because each vertex bears a one-to-one correspondence to an appropriately labeled atom in the molecule. The relationship of edges in the graph to bonds in the molecule is, however, far less well defined. This point warrants a reiteration of earlier remarks on this theme52,71, 72-79 93 because it has an important bearing on the subject of topological chirality in molecules. 

The union GiUG2 is defined as the graph which vertex set is V,UV2 and edge set is E[UE2. A disconnected graph is a graph which can be expressed as the union of two graphs [54a]... 

The concept of molecular graph was outlined in our previous communications [16,25]. This term will be generalized in such a way that some vertices are distinguished from other ones and they are called the virtual vertices. These virtual vertices are assembled in a separate virtual vertex set W = w, w, .... The remaining vertices form the vertex set (non empty) A = v, v, .... For simplicity, in the following chapter we shall not consider the description of vertices by atomic labels, but we suppose that they have a defined description. Virtual vertices have... 

The notion of isomerism for synthons was defined in [18,21,16], in the present communication we shall study the synthons that are constructed over the same vertex set A. Therefore, they are automatically isomeric. The set of all synthons (nonisomorphic) constructed over the set A is called the family of isomeric synthons, and is denoted by 3F A). The synthons from a family will be advantageously classified in our forthcoming considerations as stable, unstable, and forbidden. This will be done by making use of the concept of valence states of vertices. 

The above introduced classification scheme of synthons will be of great importance for the elaboration of effective heuristics to reduce the enormous number of synthons from the family Sr (A), where A is a given vertex set. Forbidden synthons are removed from the family "(A). 

The vertex set AR A is composed of vertices incident with edges from ER. 

Unfortunately, the mapping of virtual vertices is not unambiguous due to the fact that the virtual vertex sets W2 and W2 of Si(/l) and S2(/l), respectively, are... 

For a fixed family 3 (A) of isomeric synthons we construct the so-called graph of reaction distances [18, 21, 16, 25] denoted by RD(A). The vertex set of this graph is formally identical with the family A) without forbidden synthons, its two distinct vertices v and v, assigned to the synthons S(d) and S (/l), are connected by an edge [u, v ] if such an elementary transformation i = a, p exists so that the synthon S(/l) is transformed into the synthon i.e. 

Since the number of synthons in (S(A)) is usually enormous, we turn our attention to its reduction to a substantially smaller subset still containing all synthetically important SPS of the synthon S(A). Let X be a subset of the vertex set A, it will be called the reaction center. A subset of (S(A)) with respect to the reaction center is determined as follows. 

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