On the centrality of vertices of molecular graphs

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of “the center of a graph” by using a novel concept of graph theory, the vertex “weights,” defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex “weights,” in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices. © 2013 Wiley Periodicals, 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.     

The characteristic polynomials of structures with pending bonds

It is well known [1] that the calculation of characteristic polynomials of graphs of interest in Chemistry which are of any size is usually extremely tedious except for graphs having a vertex of degree 1. This is primarily because of numerous combinations of contributions whether they were arrived at by non-imaginative expansion of the secular determinant or by the use of some of the available graph theoretical schemes based on the enumeration of partial coverings of a graph, etc. An efficient and quite general technique is outlined here for compounds that have pending bonds (i.e., bonds which have a terminal vertex). We have extended here the step-wise pruning of pending bonds developed for acyclic structures by one of the authors [2] for elegant evaluation of the characteristic polynomials of trees by accelerating this process, treating pending group as a unit. Further, it is demonstrated that this generalized pruning technique can be applied not only to trees but to cyclic and polycyclic graphs of any size. This technique reduces the secular determinant to a considerable extent. The present technique cannot handle only polycyclic structures that have no pending bonds. However, frequently such structures can be reduced to a combination of polycyclic graphs with pending bonds [3] so that the present scheme is applicable to these structures too. Thus we have arrived at an efficient and quite a simple technique for the construction of the characteristic polynomials of graphs of any size.     

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.     

Graph generators

We consider the construction of highly symmetrical vertex transitive graphs. Some such graphs represent the degenerate rearrangements in which a molecule or an ion is formed by breaking and making bonds so that the final and the initial skeleton is identical. The approach is closely related to Cayley's graphs for selected groups. We restrict the choice of generators to symmetric matrices. Successive multiplications of such matrices generate other permutation matrices of the same dimension, each new matrix representing a new vertex for a transitive graph under the construction. In particular we restrict our discussion to matrices of dimension 3 and 4 and proceed to construct systematically all transitive graphs using 4 × 4 symmetric matrices as generators.     

Graphs whose vertices are graphs with bounded degree: Distance problems

By an f-graph we mean a graph having no vertex of degree greater than f. Let U(n,f) denote the graph whose vertex set is the set of unlabeled f-graphs of order n and such that the vertex corresponding to the graph G is adjacent to the vertex corresponding to the graph H if and only if H is obtainable from G by either the insertion or the deletion of a single edge. The distance between two graphs G and H of order n is defined as the least number of insertions and deletions of edges in G needed to obtain H. This is also the distance between two vertices in U(n,f). For simplicity, we also refer to the vertices in U(n,f) as the graphs in U(n,f). The graphs in U(n,f) are naturally grouped and ordered in levels by their number of edges. The distance nf/2 from the empty graph to an f-graph having a maximum number of edges is called the height of U(n,f). For f =2 and for f≥(n-1)/2, the diameter of U(n,f) is equal to the height. However, there are values of the parameters where the diameter exceeds the height. We present what is known about the following two problems: (1) What is the diameter of U(n,f) when 3≥f<(n-1)/2? (2) For fixed f, what is the least value of n such that the diameter of U(n,f) exceeds the height of U(n,f)?     

On Wiener‐type polynomials of thorn graphs

We derive the expressions of the ordinary, the vertex‐weighted and the doubly vertex‐weighted Wiener polynomials of a type of thorn graph, for which the number of pendant edges attached to any vertex of the underlying parent graph is a linear function of its degree. We also define variable vertex‐weighted Wiener polynomials and calculate them for the same type of thorn graphs. Copyright © 2009 John Wiley & Sons, Ltd.     

On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.     

A novel approach to the use of graph theory in structure–activity relationship studies. Application to the qualitative evaluation of mutagenicity in a series of nonfused ring aromatic compounds

In this article we describe a novel approach to the application of graph theory in structure–activity relationship studies. An information–theoretical topological index for the vertices of a molecular graph has been used for the qualitative evaluation of the mutagenic activity of a series of nonfused ring aromatic compounds. The use of a vertex index contrasts with the conventional approach of using a topological index for the entire molecule. The idea is to identify regions, or substructures in the molecules (molecular graphs) which may be used to determine certain biological activity of chemical compounds. The results obtained in this paper indicate that the present approach is capable of classifying the mutagenic activity of the compounds under consideration and may find useful application in structure–activity relationship studies of diverse bioactive compounds.     

Generalized graphs and the Sinanoğlu graphical rules

A comparison of Sinano?lu's VIF (Ref. 1) and generalized graph is presented. Generalized graphs have vertex and edge weights. An abridged history of generalized graphs in theoretical chemistry is given. VIF 's are generalized graphs and therefore have adjacency matrices. The “graphical” rules of Sinano?lu can be represented by congruent transformations on the adjacency matrix. Thus the method of Sinano?lu is incorporated into the broad scheme of graph spectral theory. If the signature of a graph is defined as the collection of the number of positive, zero, and negative eigenvalues of the graph's adjacency matrix, then it is identical to the all-important {n₊, n₀, n_?}, the {number of positive, zero, and negative loops of a reduced graph} or the {number of bonding, nonbonding, and antibonding MO s}. A special case of the Sinano?lu rules is the “multiplication of a vertex” by (?1). In matrix language, this multiplication is an orthogonal transformation of the adjacency matrix. Thus, one can multiply any vertex of a generalized graph by ?1 without changing its eigenvalues.     

密度泛函活性理论中的Rényi熵, Tsallis熵和Onicescu信息能

根据密度泛函理论,分子的电子密度确定了该体系基态下的所有性质,其中包括结构和反应活性.如何运用电子密度泛函有效地预测分子反应活性仍然是一个有待解决的难题.密度泛函活性理论(DFRT)倾力打造这样一个理论和概念架构,使得运用电子密度以及相关变量准确地预测分子的反应特性成为可能.信息理论方法的香农熵和费舍尔信息就是这样的密度泛函,研究表明,它们均可作为反应活性的有效描述符.本文将在DFRT框架中介绍和引进三个密切相关的描述符, Rényi熵、Tsallis熵和Onicescu信息能.我们准确地计算了它们在一些中性原子和分子中的数值并讨论了它们随电子数量和电子总能量的变化规律.此外,以第二阶Onicescu信息能为例,在分子和分子中的原子两个层面上,系统地考察了其随乙烷二面角旋转的变化模式.这些新慨念的引入将为我们深入洞察和预测分子的结构和反应活性提供额外的描述工具.     

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.     

On reciprocal reverse Wiener index

We report some properties, especially bounds for the reciprocal reverse Wiener index of a connected (molecular) graph. We find that the reciprocal reverse Wiener index possesses the minimum values for the complete graph in the class of n-vertex connected graphs and for the star in the class of n-vertex trees, and the maximum values for the complete graph with one edge deleted in the class of n-vertex connected graphs and for the tree obtained by attaching a pendant vertex to a pendant vertex of the star on n − 1 vertices in the class of n-vertex trees. These results are compared with those obtained for the ordinary Wiener index.     

Bounds on Harary index

In this paper, we obtain the lower and upper bounds on the Harary index of a connected graph (molecular graph), and, in particular, of a triangle- and quadrangle-free graphs in terms of the number of vertices, the number of edges and the diameter. We give the Nordhaus–Gaddum-type result for Harary index using the diameters of the graph and its complement. Moreover, we compare Harary index and reciprocal complementary Wiener number for graphs.     

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.     

Three-dimensional chiral objects and their star graph representations

Planar chirality properties can be analysed using n-polyominoes and graphs. In this paper we study graph representations of three-dimensional chiral objects and discuss the generalization of planar case. We show that graph representations of three-dimensional chiral objects can be star graphs.     

On the ABC index of connected graphs with given degree sequences

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.     

Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks

Using probabilistic tools, we give tight upper and lower bounds for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and the spectral gap of the transition probability matrix associated to the random walk on the graph. We then use bounds of the spectral gap of more specialized graphs, available in the literature, in order to obtain upper bounds for the Kirchhoff index of these specialized graphs. As a byproduct, we obtain a closed‐form formula for the Kirchhoff index of the d‐dimensional cube in terms of the first inverse moment of a positive binomial variable. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

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.