Labeled molecular graph

John, P.E., Malhon, R.B. and Gutman, I. (1998). An Algorithm for Counting Spanning Trees in Labeled Molecular Graphs Homeomorphic to Cata-Condensed Systems. J.Chem.Inf Corn-put. ScL, 38,108-112. 

The Hiickel Hamiltonian-matrix for an n-annulene (Fig. 5-1) is the matrix in the determinant found in equation (5-1), with the quantities ( — x), along the diagonal, replaced by zero (i.e. a). As pointed out in Appendix A, this matrix is isomorphic with the adjacency matrix, A(C ), of the correspondingly-labelled molecular graph furthermore, the eigenvalues of the matrix, A(C ), are the annulene energy-levels we require (expressed in units of fi and... 

The labeled molecular graph underlying a molecule with n atoms is a connected... 

The molecule generator MOLGEN, up to version 3.5 [19], generates compounds on the level of RC or CSC (if we enter particular valences, e.g. 5 for P, or 4 or 6 for S), while from version 4.0 onwards [148] it is possible to go to level 1C. Molecules (compounds) generated are unlabeled molecular graphs, i.e. equivalence classes of labeled molecular graphs that we introduce as follows ... 

Definition (Labeled molecular graphs of n atoms in ) Let n denote the number of atoms in the molecular graphs to be defined. Consider a set of chemical elements, and denote by... 

If y 9, then the orbit S ((e, y)) e S lvt of the labeled molecular graph (e, y) is called a connected unlabeled molecular graph. Equivalence classes of connected labeled molecular graphs can be identified with constitutional formulas of chemical compounds. Exceptions are exotic compounds such as catenanes and rotaxanes, whose structural formulas correspond to disconnected graphs. These are not dealt with in this book. The notation is... 

The orbit S (M) of M is denoted by M. We denote the set of the labeled molecular graphs with n atoms by M euid use this notation to introduce the following sets ... 

We recall from Section 2.6 that a molecular descriptor D comes from a mapping D on the set of labeled molecular graphs that is invariant under relabeling of a graph s nodes. In formal terms, if M = (e, C, y) e M , we require that D M) = D nM), for each relabeling n S . This gives rise to two molecular descriptors D and D with the values... 

As D neglects hydrogen atoms, it is helpful to introduce the following notation for the non-H atoms in a given labeled molecular graph M = (e, y) M ,... 

I. Gutman, R.B. Malbon, and J.W. Essam, Counting spanning trees of a labelled molecular-graph, Mol Phys. 50 (1983) 859-877. 

We begin with the way chemists perceive similarity between two molecules. This process involves, consciously or unconsciously, comparing several types of structural features present in the molecules. For example, considering the five aliphatic alcohols (represented by their H-suppressed molecular graphs) in Figure 1, we note both similarities and differences they are all four-carbon alcohols a, b, c and d are acyclic, whereas e has a ring a and b are primary alcohols, c and e are secondary alcohols and d is a tertiary alcohol b and c have the same skeleton, but for the labeling of points (atoms), while the other skeletons are distinct etc. 

Figure 3.11 Contour lines (gray) of the electron density, the molecular graph (black), and interatomic surfaces (black) in the dimeric structure (BH3-NH3)2 obtained at the MP2/6-31G level. Here the bond critical points are marked as squares and the ring critical points as triangles. The labels of the nuclei located in the mirror plane (the plane of the paper) are solid, and those that do not lie in this plane are open. (Reproduced with permission from ref. 16.)...

Figure 6.13 The DIOP ligand a structure b molecular graph c adjacency matrix. The broken linesindicatetheshortestand longest P-P connectivity paths, Dpi p2 and AP1 P2, respectively. The adjacency matrix of a molecular graph is a matrix with rows and columns labeled by graph vertices v (i.e., the atoms), with a 1 or 0 in position (vp Vj) according to whether Vj or Vj are adjacent or not.

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. 

D structures of drugs were converted into hydrogen-suppressed molecular graphs, where nodes are labeled with atom types except hydrogens and edges are labeled with bond types. 

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 value of the charge density at a bond critical point can be used to define a bond order (Bader et al. 1983 Cremer and Kraka 1984). The molecular graphs for ethane, ethylene, and acetylene are shown in Fig. 2.8. In each case the unique pair of trajectories associated with a single (3, — 1) critical point is found to link the carbon nuclei to one another. Multiple bonds do not appear as such in the topology of the charge density. Instead, one finds that the extent of charge accumulation between the nuclei increases with the assumed number of electron pair bonds and this increase is faithfully monitored by the value of p at the bond critical point, a value labelled p, . For carbon-carbon bonds, one can define a bond order n in terms of the values of Ph using a relationship of the form... 

Fig. 2. 2D representation (molecular graph) of icosahedral Qo, with faces labelled according to an orange peel scheme. The pentagonal face labelled 32 lies on the opposite side of the molecule to the pentagonal face labelled 1. 

Given a -+ molecular graph ( , where vertices are labelled by the chemical element of the corresponding atoms, cluster expansion in the additive form is among the group contribution methods expressing a molecular property 0 as a sum of contributions of all the connected subgraphs of Q, i.e. 

Faulon, J.-L. (1998). Isomorphism, Automorphism Partitioning, and Canonical Labeling Can Be Solved in Polynomial-Time for Molecular Graphs. J.Chem.lnf.Comput.Scl, 38,432-444. 

Labeled Hydrogen-Filled molecular Graph = H-jiUed molecular graph molecular graph... 

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