Equivalent molecular graphs

Fig. 2-9 Structural formula of ethylene and its equivalent molecular-graph representing the ir-bond connectivity of its carbon atoms.

Two molecular graphs M, M )vt are equivalent molecular graphs or isomorphic molecular prap/isiftheybelongtothesameorbitunderthis action. The elements of = S )vt , i.e. the orbits, are called equivalence classes of molecular graphs. The class of M will be indicated as M, and the set of all these classes is denoted by... 

Molecular structure Molecular structure Equivalent class of molecular graphs... 

Figure 12.7 presents the molecular graph of CLOH—HBeH complex where the intermolecular O-H—H-Be DHB exists. It was found that this structure corresponds to the stable energetic minimum [35], This is the shortest ever found theoretically intermolecular contact (H—H) since it amounts to only 1.049 A. On the other hand, such a short contact represents the proton-acceptor distance.9 It is worth mentioning that the other shortest contact was revealed for bifluoride (FHF)-ion. For this system, the H—F equivalent distances amount to 1.1—1.2 A. The exact value depends on the type of experimental measurements or the applied method of calculations [36]. 

Identification of all the molecular structures compatible with (r,R,t), that is construction of a set (]j> of all distinct equivalence classes of <7(R, t)>. Two molecular graphs belong to the same equivalence class Qj if and only if they are homeomorphic [27,28]. The set Qj is finite, provided that the system under consideration consists of a finite number of particles (nuclei and electrons). 

In Table 2 we illustrate equivalence classes for some selected sets modeled by alkane molecular graphs [lO]-... 

Fig. 21 A grid resulting when the Randic -Wilkins partial ordering conditions is adopted for the isomers of octanes The values next to the vertices are mean information-theoretic indices, I, on the permutation electronic integrals of the equivalent polyhexes Starred positions indicate cases where the molecular graph of a given octane is not a caterpillar and hence for which a polyhex is not defined The template leads to excellent partial ordering where J values always increase from left to right-...

An equivalence relation for molecular graphs is defined as follows two molecular graphs are equivalent if and only if they are associated with two points of the same structural region. An equivalence class of molecular graphs is called a molecular structure. It is then seen that a unique molecular... 

Step 1. Vertices of the -> H-depleted molecular graph are partitioned into equivalence classes according to their vertex degree 6, i.e. a first rank = 6, is assigned to each vertex. 

Topological information indices are graph invariants, based on information theory and calculated as - information content of specified equivalence relationships on the molecular graph. 

Hall, L.H. and Kier, L.B. (1990). Determination of Topological Equivalence in Molecular Graphs from the Topological State. Quant.Struct.-Act.Relat., 9,115-131. 

---