New vertex invariants and topological indices of chemical graphs based on information on distances

By applying information theory to the set of topological distances from one vertex to all other graph vertices, one obtains four new types of vertex invariants (u _i,v _i,x _i,Y _i) which are real numbers (as opposed to integers). They may be combined in many ways to afford new topological indices. One such type leads to indicesU, V, X andY which show no degeneracy for alkanes with up to 15 vertices.     

Computer generation of acyclic graphs based on local vertex invariants and topological indices. Derived canonical labelling and coding of trees and alkanes

A new procedure (GENLOIS) is presented for generating trees or certain classes of trees such as 4-trees (graphs representing alkanes), identity trees, homeomorphical irreducible trees, rooted trees, trees labelled on a certain vertex (primary, secondary, tertiary, etc.). The present method differs from previous procedures by differentiating among the vertices of a given parent graph by means of local vertex invariants (LOVIs). New graphs are efficiently generated by adding points and/or edges only to non-equivalent vertices of the parent graph. Redundant generation of graphs is minimized and checked by means of highly discriminating, recently devised topological indices based either on LOVIs or on the information content of LOVIs. All trees onN + 1 (N + 1 < 17) points could thus be generated from the complete set of trees onN points. A unique cooperative labelling for trees results as a consequence of the generation scheme. This labelling can be translated into a code for which canonical rules were recently stated by A.T. Balaban. This coding appears to be one of the best procedures for encoding, retrieving or ordering the molecular structure of trees (or alkanes).Dedicated to Professor Alexandru T. Balaban on the occasion of his 60th anniversary.     

Search for nondegenerate real vertex invariants and derived topological indexes

Two new types of real (i.e., noninteger) local vertex invariants (LOVIs), denoted by c_i and c′_i and called distance-enhanced exponential connectivities, are defined via eqs. (1)–(3) and (1′)–(3′), respectively. Only the case when the exponent z equals 1 in eqs. (3) and (3′) is discussed in detail. Both these LOVIs span the range from 0–1, but their dependence upon topology is fairly different, as seen from Table II, where c_i and c′_i values for all heptane and octane isomers are displayed. From these LOVIs, by simple summation over all graph vertices two new topological indexes (TIs), denoted by XC and XC′, respectively, are obtained. Their intermolecular ordering of all alkanes with four to nine carbon atoms is discussed. On their basis, correlations with boiling points and critical pressures of alkanes are presented. © 1993 John Wiley & Sons, Inc.     

Resistance distance local rules

In [D.J. Klein, Croat. Chem. Acta. 75(2), 633 (2002)] Klein established a number of sum rules to compute the resistance distance of an arbitrary graph, especially he gave a specific set of local sum rules that determined all resistance distances of a graph (saying the set of local sum rules is complete). Inspired by this result, we give another complete set of local rules, which is simple and also efficient, especially for distance-regular graphs. Finally some applications to chemical graphs (for example the Platonic solids as well as their vertex truncations, which include the graph of Buckminsterfullerene and the graph of boron nitride hetero-fullerenoid B ₁₂ N ₁₂) are made to illustrate our approach.     

Chemical graphs with degenerate topological indices based on information on distances

Recently, four new types of vertex invariants, namelyu, v, x, andy, were defined on the basis of information on graph distances. They were combined to give four highly selective topological indices:U, V, X, andY. The degeneracy, i.e. equal values for nonisomorphic graphs, of the four topological indices is investigated. A structural condition and a graphical method which gives pairs of molecular graphs with identicalU, V, X, andY topological indices are introduced. The smallest pair of 4-trees representing alkanes having degeneratedU, V, X, andY values consists of trees with eighteen vertices.     

Characterization of molecular complexity

We put forward a novel index of molecular complexity, ξ, taking into account the symmetry of a molecular graph and the specificity of structural components considered. The ξ index is defined as the sum of augmented valences of all mutually nonequivalent vertices in a molecular graph. The augmented valence of a vertex in a graph is the sum of its valence and valences of all neighboring vertices with the weight 1/2^d depending on their distance, d, from the vertex. The ξ index is examined on the set of octane isomers and some special classes of graphs. It is also compared with a certain number of alternative complexity measures considered in the literature. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

Molecular topology. IV. Regressive vertex degrees (new graph invariants) and derived topological indices

New local graph invariants, “regressive vertex degrees” (which are slightly augmented vertex degrees) are introduced on the basis of decreasing contributions of more remote vertexes to the classical vertex degrees. Several such invariants are proposed (BR, ER, SR) where t (either t = 1 or t = 2) is an operator expressing the attenuation with increasing topological distance, according to formula (1) or (2). With the aid of these new local invariants, new topological indices (global graph invariants), Y (namely BY, EY or SY) are introduced and exemplified. Their ability to express the branching and to order alkanes is investigated. An appendix gives some recursive relationships for computing these indices.     

Novel Indices for the Topological Complexity of Molecules

Abstract

The development of molecular complexity measures is reviewed. Two novel sets of indices termed topological complexities are introduced proceeding from the idea that topological complexity increases with the overall connectivity of the molecular graph. The latter is assessed as the connectivity of all connected subgraphs in the molecular graph, including the graph itself. First-order, second-order, third-order, etc., topological complexities ⁱ TC are defined as the sum of the vertex degrees in the connected subgraphs with one, two, three, etc., edges, respectively. Zero-order complexity is also specified for the simplest subgraphs–the graph vertices. The overall topological complexity TC is then defined as the sum of the complexities of all orders. These new indices mirror the increase in complexity with the increase in the number of atoms and, at a constant number of atoms, with the increase in molecular branching and cyclicity. Topological complexities compare favorably to molecular connectivities of Kier and Hall, as demonstrated in detail for the classical QSPR test-the boiling points of alkanes. Related to the wide application of molecular connectivities to QSAR studies, a similar importance of the new indices is anticipated.     

Topological indices for molecular fragments and new graph invariants

Whereas the internal fragment topological index (IFTI) is calculated in the normal manner as for any molecule, the external fragment topological index (EFTI) is calculated so as to reflect the interaction between the excised fragment F and the remainder of the molecule (G-F). For selected topological indices (TIs), a survey of EFTI values, formulas and examples is presented. Some requirements as to the fragment indices are formulated and examined. In the discussion of the results, it is shown that for some TIs regularities exist in the dependence of EFTI values upon the branching of fragment F, or upon the marginal versus central position of the fragment F in the graph G. New vortex invariants can be computed as EFTI values for one-atom fragments over all graph vertices; by iteration, it is in principle possible to devise an infinite number of now vertex invariants.     

The graph center concept for polycyclic graphs

While the concept of the graph center is unambiguous (and quite old) in the case of acyclic graphs, an attempt has been made recently to extend the concept to polycyclic structures using the distance matrix of a graph as the basis. In this work we continue exploring such generalizations considering in addition to the distance matrix, self-avoiding walks or paths as graph invariants of potential interest for discriminating distinctive vertex environments in a graph of polycyclic structures. A hierachy of criteria is suggested that offers a systematic approach to the vertex discrimination and eventually establishes in most cases the graph center as a single vertex, a single bond (edge), or a single group of equivalent vertices. Some applications and the significance of the concept of the graph center are presented.     

Unique description of chemical structures based on hierarchically ordered extended connectivities (HOC procedures). V. New topological indices,ordering of graphs,and recognition of graph similarity

The vertex numbering obtained by application of the HOC algorithm can be converted into two sequences of numbers: If each vertex starting with vertex 1 is only counted once, the sums of numberings of adjacent vertices form sequence S_i (i = 1?N), while the sums of S_i values form sequence M_i (i = 1?N). These two sequences can be used for (i) two new topological indices, ?? and ??, the latter being of extremely low degeneracy, and the former correlating with boiling points of alkanes; (ii) a criterion based on sequence S_i for ordering graphs which possess the same number N of vertices; and (iii) a quantitative measure, also based on sequence S_i, for appreciating the similarity or dissimilarity of pairs of graphs. Comparisons with other topological indices, ordering criteria, and similarity measures for graphs show that the newly devised procedures compare favorably with those known previously.     

Computer generation of the characteristic polynomials of chemical graphs

A computer program based on the Frame method for the characteristic polynomials of graphs is developed. This program makes use of an efficient polynomial algorithm of Frame for generating the coefficients in the characteristic polynomials of graphs. This program requires as input only the set of vertices that are neighbors of a given vertex and with labels smaller than the label of that vertex. The program generates and stores only the lower triangle of the adjacency matrix in canonical ordering in a one-dimensional array. The program is written in integer arithmetic, and it can be easily modified to real arithmetic. The coefficients in the characteristic polynomials of several graphs were generated in less than a few seconds, thus solving the difficult problem of generating characteristic polynomials of graphs. The characteristic polynomials of a number of very complicated graphs are obtained including for the first time the characteristic polynomial of an honeycomb lattice graph containing 54 vertices.     

Intrinsic local constituents of molecular electronic wave functions. I. Exact representation of the density matrix in terms of chemically deformed and oriented atomic minimal basis set orbitals

A coherent, intrinsic, basis-set-independent analysis is developed for the invariants of the first-order density matrix of an accurate molecular electronic wavefunction. From the hierarchical ordering of the natural orbitals, the zeroth-order orbital space is deduced, which generates the zeroth-order wavefunction, typically an MCSCF function in the full valence space. It is shown that intrinsically embedded in such wavefunctions are elements that are local in bond regions and elements that are local in atomic regions. Basis-set-independent methods are given that extract and exhibit the intrinsic bond orbitals and the intrinsic minimal-basis quasi-atomic orbitals in terms of which the wavefunction can be exactly constructed. The quasi-atomic orbitals are furthermore oriented by a basis-set independent method (viz. maximization of the sum of the fourth powers of all off-diagonal density matrix elements) so as to exhibit clearly the chemical interactions. The unbiased nature of the method allows for the adaptation of the localized and directed orbitals to changing geometries. Contribution of the Mark S. Gordon 65th Birthday Fegtschrift Issue.