Atomic walk counts of negative order

Atomic walk counts (awc's) of order k (k > or = 1) are the number of all possible walks of length k which start at a specified vertex (atom) i and end at any vertex j separated by m (0 < or = m < or = k) edges from vertex i. The sum of atomic walk counts of order k is the molecular walk count (mwc) of order k. The concept of atomic and molecular walk counts was extended to zero and negative orders by using a backward algorithm based on the usual procedure used to obtain the values of mwc's. The procedure can also be used in cases in which the adjacency matrix A related to the actual structure is singular and therefore A(-1) does not exist. awc's and mwc's of negative order may assume noninteger and even negative values. If matrix A is singular, atomic walk counts of zero order may not be equal to one.     

Method for construction of characteristic polynomials via graph linearization

A new method for construction of characteristic polynomials (CP) of complicated graphs having arbitrary edge and vertex weights has been developed. The method first converts the graph into isospectral linear chains with weighted vertices and edges and then builds up the CP coefficients recursively. Two types of graphs have been used for illustration, viz., (i) graphs that can be linearized by symmetry factorization and (ii) graphs without symmetry which are to be linearized by an algorithm involving walks of unit length. Both types have been illustrated, of which type (i) includes the Schlegel of fullerene fragment C₂₀ and another large graph with many fused rings. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 199–204, 1997     

Computer enumeration of walks on directed graphs

A vectorized computer code is developed for the enumeration of walks through the matrix power method for directed graphs. Application of this code to several graphs is considered. It is shown that the coefficients in the generating functions for signed graphs are much smaller in magnitude. It is shown that self-avoiding walks on some graphs can be enumerated as a linear combination of walk GFs of directed paths and rooted-directed paths.     

Novel shape descriptors for molecular graphs.

We report on novel graph theoretical indices which are sensitive to the shapes of molecular graphs. In contrast to the Kier's kappa shape indices which were based on a comparison of a molecular graph with graphs representing the extreme shapes, the linear graph and the "star" graph, the new shape indices are obtained by considering for all atoms the number of paths and the number of walks within a graph and then making the quotients of the number of paths and the number of walks the same length. The new shape indices show much higher discrimination among isomers when compared to the kappa shape indices. We report the new shape indices for smaller alkanes and several cyclic structures and illustrate their use in structure-property correlations. The new indices offer regressions of high quality for diverse physicochemical properties of octanes. They also have lead to a novel classification of physicochemical properties of alkanes.     

A model for combinatorial organic chemistry

The set of coset representations, CR's, of a group G, [G(/G1), G(/G2), ..., G(/Gs)], where G1 = [I], Gs = G, the marks, m(ij) of subgroup Gj on a given G(/Gi), 1 < or = i < or = s, and the subduction of G(/Gi) by Gj, j < or = i, G(/Gi) precipitates to Gj, are essential tools for the enumeration of stereoisomers and their classification according to their subgroup symmetry (Fujita, S. Symmetry and Combinatorial Enumeration in Chemistry; Springer-Verlag: Berlin, 1991). In this paper, each G(/Gi) is modeled by a set of colored equivalent configurations (called homomers), H = h1, h2, ..., hr, r = G/Gi, such that a given homomer, h(k), remains invariant only under all g epsilon Gi, where g is an element of symmetry. The resulting homomers generate the corresponding set of marks almost by inspection. The symmetry relations among a set H can be conveniently stored in a Cayley-like diagram (Chartrand, G. Graphs as Mathematical Models; Prindle, Weber and Schmidt Incorporated: Boston, MA, 1977; Chapter 10), which is a complete digraph on r vertices so that an arc from vertex v(i) to vertex v(j) is colored with the set Sij of symmetry elements such that h(i)[g(if)]-->h(j),g(ij) epsilon Sij. In addition, each vertex, v(i), is associated with a loop that is colored with a set Sii so that g(ii) epsilon Sii stabilizes h(i). A Cayley-like diagram of a given CR, G[G(/Gi)], leads to graphical generation of G(/Gi) precipitates to Gj for all values of j and also to all m(ij)'s. Several group-theoretical results are rederived and/or became more envisagable through this modeling. The approach is examplified using C2, C3, D2, T, and D3 point groups and is applied to trishomocubane, a molecule that belongs to the D3 point group.     

Expected distance based on random walks

By considering a graph as a network of resistances, Klein and Randi? (J Math Chem 12(1):81–95, 1993) proposed the definition of a distance measure. Indeed, if each edge of the graph represents a resistance of \(1 \varOmega \), the equivalent resistance of the graph between each pair of vertices may be used as a distance. Based upon random walks in graphs, Stephenson and Zelen (Soc Netw 11(1):1–37, 1989) built a computational model to find the probability that each edge is used. From a mathematical point of view, both articles are based upon exactly the same model and the link between random walks and the electrical representation was established by Newman (Soc Netw 27(1):39–54, 2005) when defining an alternative to Freeman’s (Sociometry 40:35–41, 1977, Soc Netw 1(3):215–239, 1979) betweenness centrality based upon random walks. In the present paper, the similitude between these two processes is exploited to propose a new random walks based distance measure that may be defined as the expected length of a walk between any pair of vertices. We call it the expected distance and we prove that it is actually a distance. From this new definition, the RW Index is proposed that sums the expected walks lengths between pairs of vertices exactly in the same way as the Wiener index sums the shortest paths distances or the Kirchhoff index sums the equivalent resistances. We compare the three indices and establish the vertex and the edge decompositions for both. We compute some value of the RW index for some families of graphs and conjecture the upper and lower bounds of the RW index.     

A kinetic model of the formation of organic monolayers on hydrogen-terminated silicon by hydrosilation of alkenes

We have analyzed a kinetic model for the formation of organic monolayers based on a previously suggested free radical chain mechanism for the reaction of unsaturated molecules with hydrogen-terminated silicon surfaces (Linford, M. R.; Fenter, P. M.; Chidsey, C. E. D. J. Am. Chem. Soc 1995, 117, 3145). A direct consequence of this mechanism is the nonexponential growth of the monolayer, and this has been observed spectroscopically. In the model, the initiation of silyl radicals on the surface is pseudo first order with rate constant, ki, and the rate of propagation is determined by the concentration of radicals and unreacted Si-H nearest neighbor sites with a rate constant, kp. This propagation step determines the rate at which the monolayer forms by addition of alkene molecules to form a track of molecules that constitute a self-avoiding random walk on the surface. The initiation step describes how frequently new random walks commence. A termination step by which the radicals are destroyed is also included. The solution of the kinetic equations yields the fraction of alkylated surface sites and the mean length of the random walks as a function of time. In mean-field approximation we show that (1) the average length of the random walk is proportional to (kp/ki)1/2, (2) the monolayer surface coverage grows exponentially only after an induction period, (3) the effective first-order rate constant describing the growth of the monolayer and the induction period (kt) is k = (2ki kp)1/2, (4) at long times the effective first-order rate constant drops to ki, and (5) the overall activation energy for the growth kinetics is the mean of the activation energies for the initiation and propagation steps. Monte Carlo simulations of the mechanism produce qualitatively similar kinetic plots, but the mean random walk length (and effective rate constant) is overestimated by the mean field approximation and when kp > ki, we find k approximately ki0.7kp0.3 and Ea = (0.7Ei+ 0.3Ep). However the most striking prediction of the Monte Carlo simulations is that at long times, t > 1/k, the effective first-order rate constant decreases to ki even in the absence of a chemical termination step. Experimental kinetic data for the reaction of undec-1-ene with hydrogen-terminated porous silicon under thermal reflux in toluene and ethylbenzene gave a value of k = 0.06 min(-1) and an activation energy of 107 kJ mol(-1). The activation energy is in reasonable agreement with density functional calculations of the transition state energies for the initiation and propagation steps.     

Random walks and their diagnostic value for characterization of atomic environment

A characterization of atomic environments based on counting random walks in a molecular skeleton is outlined. To each atom in a molecule a sequence of integers w₁, w₂, w₃,…, w_n is assigned, where w_i represents the number of self-returning walks of length k, the length being defined by the number of bonds traversed. Properties of the derived atom codes are discussed. The codes display an impressive diversity and are superior to atomic codes based on enumeration of self-avoiding walks (or paths) in discriminating atomic environments. In certain cases the codes of individual atoms are not unique and the same codes appear in different molecules or even within the same molecule. The occurrence of the nonunique codes can be related to special structural situations, associated with the occurrence of isospectral graphs. These isospectral graphs which have atoms with identical codes can generate additional isospectral structures by attaching any arbitrary group to such points. If nonequivalent atoms of a single molecule have identical random walk codes, substitution at the singular points alternatively will produce isospectral graphs. Examples of such situations are given.     

G-quadruplex DNA assemblies: loop length, cation identity, and multimer formation

G-rich DNA sequences are able to fold into structures called G-quadruplexes. To obtain general trends in the influence of loop length on the structure and stability of G-quadruplex structures, we studied oligodeoxynucleotides with random bases in the loops. Sequences studied are dGGGW(i)GGGW(j)GGGW(k)GGG, with W = thymine or adenine with equal probability, and i, j, and k comprised between 1 and 4. All were studied by circular dichroism, native gel electrophoresis, UV-monitored thermal denaturation, and electrospray mass spectrometry, in the presence of 150 mM potassium, sodium, or ammonium cations. Parallel conformations are favored by sequences with short loops, but we also found that sequences with short loops form very stable multimeric quadruplexes, even at low strand concentration. Mass spectrometry reveals the formation of dimers and trimers. When the loop length increases, preferred quadruplex conformations tend to be more intramolecular and antiparallel. The nature of the cation also has an influence on the adopted structures, with K(+) inducing more parallel multimers than NH4(+) and Na(+). Structural possibilities are discussed for the new quadruplex higher-order assemblies.     

Scattering resonances in slow NH3-He collisions

We theoretically study slow collisions of NH(3) molecules with He atoms, where we focus in particular on the observation of scattering resonances. We calculate state-to-state integral and differential cross sections for collision energies ranging from 10(-4) cm(-1) to 130 cm(-1), using fully converged quantum close-coupling calculations. To describe the interaction between the NH(3) molecules and the He atoms, we present a four-dimensional potential energy surface, based on an accurate fit of 4180 ab initio points. Prior to collision, we consider the ammonia molecules to be in their antisymmetric umbrella state with angular momentum j = 1 and projection k = 1, which is a suitable state for Stark deceleration. We find pronounced shape and Feshbach resonances, especially for inelastic collisions into the symmetric umbrella state with j = k = 1. We analyze the observed resonant structures in detail by looking at scattering wavefunctions, phase shifts, and lifetimes. Finally, we discuss the prospects for observing the predicted scattering resonances in future crossed molecular beam experiments with a Stark-decelerated NH(3) beam.     

Walk counts, labyrinthicity, and complexity of acyclic and cyclic graphs and molecules

It is demonstrated how the complexity of a (molecular) graph can be quantified in terms of the walk counts, extremely easily obtained graph invariants that depend on size, branching, cyclicity, and edge and vertex weights (unsaturation, heteroatoms). The influence of symmetry is easily accounted for. The term labyrinthicity is proposed for what is measured by walk counts alone, neglecting symmetry. The total walk count and recently advanced measures of labyrinthicity or complexity are compared with respect to the ordering of structures and to the computational effort required to obtain numerical values.     

定量的微扰晶体轨道法

我们将研究小分子的结构与性能关系的定量微扰分子轨道法(定量PMO法)推广到晶体(或高分子)中,在从头算自洽场晶体轨道(SCFCO)基础上,提出了一个定量的微扰晶体轨道法并编制了计算机程序对几个高分子进行了计算。这种方法按照问题的需要,把晶胞(或称单胞)分成两个片断晶胞。然后求出片断从头算自洽场晶体轨道;利用微扰理论,可求得片断晶体轨道间的相互作用能,并用这种相互作用定量地对晶体或高分子性能的影响进行解释。     

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)?     

Error-systematics of determining simultaneously the isotopic abundance ratios of natural lithium and natural boron as Li2BO2+

Recently, we have shown how the errors delta(j) and delta(k), that occur when measuring the two different isotopic molecular abundance ratios required for analysis, are transformed into the actual errors of elemental isotopic analysis, (deltaEi/Ei)'s. With a view to gain further understanding as to how the errors (deltaEi/Ei)'s are governed, we now evaluate theoretically the effects of selecting different isotopic molecular pairs as the monitor pairs (j and k) for measurements, and of the measurement errors (delta(j) and delta(k)), on the results of analysis (the 6Li/7Li and the 10B/11B abundance ratios), by considering all the constituent elements of Li2BO2+ at their natural isotopic abundances. It is shown that the ratio of measurement errors, delta(j)/delta(k), is a more fundamental parameter than either the individual errors (delta(j) and delta(k)), or their sum, absolute value(delta(j)) + absolute value(delta(k)), in governing deltaEi/Ei. The important implication of this observation is that it reveals the possibility of achieving not only a desired level of accuracy in analysis, but even absolute accuracy (i.e. deltaEi/Ei = 0) by causing mutual cancellation of the effects of individual measurement errors delta(j) and delta(k), through proper regulation of measurement parameters. However, as the measurement errors cannot be pre-set, it is shown how selection of proper monitor pairs (j and k) can help achieve the desired accuracy in analysis. The present work sets guidelines for the more general problem of selecting monitor pairs to avoid larger errors in analysis.     

π和σ体系彻底分离的高度定域的键轨道基组的建立

高度定域的、对称的、键轨道基组的建立是一个多步的计算程序：(1)以定域片断轨道[φk,φi,φj]为基,对分子作有条件的RHF运算,算得FUL和DSI°态的片断分子轨道[Φ°l',Φ°n,Φ°m]和[Φl,Φn,Φm]。在基组[φk,φi,φj]中,φi∈双占据和空σ片断分子轨道(FMOs)组,φj∈πFMO组,φk∈单占据σFMO组,它们都精确地定域在各自的片断内;(2)利用Φ°l'与Φ°l间的重叠积分值(Sl'l>0.5),可以从DSI°态中,自动地选出Ns个对称的、由单占据轨道线性组合而成的分子轨道Φ°l'=Σakl'φk(k=1,2,…,Ns),接着,用Φ°l'取代FUL态中同类的、非对称轨道组Φl=Σaklφk(k=1,2,…,Ns);(3)以上述新的轨道组[Φ°l',Φn,Φm]为基(其中,Φ°l'∈DSI°态,它们离域于整个分子;双占据及空σFMO组Φm和πFMO组Φm属于FUL态),按FUL态的条件,再次对分子作有条件的RHF运算,从中得到一组对称的、闭壳层正则FMOs,而且每一个FMO均有正确的电子占据数;(4)利用Perkin原理,将第3步所得的正则FMO组定域成一个对称的键轨道基组[Φl',Φn',Φm']。在这个基组中,π体系Φm'与σ构架Φn'是彻底分离的,而且这两个轨道组始终精确地定域在各自的片断内。     

Four new topological indices based on the molecular path code

The sequence of all paths pi of lengths i = 1 to the maximum possible length in a hydrogen-depleted molecular graph (which sequence is also called the molecular path code) contains significant information on the molecular topology, and as such it is a reasonable choice to be selected as the basis of topological indices (TIs). Four new (or five partly new) TIs with progressively improved performance (judged by correctly reflecting branching, centricity, and cyclicity of graphs, ordering of alkanes, and low degeneracy) have been explored. (i) By summing the squares of all numbers in the sequence one obtains Sigmaipi(2), and by dividing this sum by one plus the cyclomatic number, a Quadratic TI is obtained: Q = Sigmaipi(2)/(mu+1). (ii) On summing the Square roots of all numbers in the sequence one obtains Sigmaipi(1/2), and by dividing this sum by one plus the cyclomatic number, the TI denoted by S is obtained: S = Sigmaipi(1/2)/(mu+1). (iii) On dividing terms in this sum by the corresponding topological distances, one obtains the Distance-reduced index D = Sigmai{pi(1/2)/[i(mu+1)]}. Two similar formulas define the next two indices, the first one with no square roots: (iv) distance-Attenuated index: A = Sigmai{pi/[i(mu + 1)]}; and (v) the last TI with two square roots: Path-count index: P = Sigmai{pi(1/2)/[i(1/2)(mu + 1)]}. These five TIs are compared for their degeneracy, ordering of alkanes, and performance in QSPR (for all alkanes with 3-12 carbon atoms and for all possible chemical cyclic or acyclic graphs with 4-6 carbon atoms) in correlations with six physical properties and one chemical property.     

Nonisomorphic graphs with identical atomic counts of self-returning walks: Isocodal graphs

On the basis of endospectral graphs, we present a graphical method for obtaining pairs of nonisomorphic graphs with identical atomic counts of self-returning walks.     

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.     

Unicycle graphs with extremal Merrifield–Simmons Index

The Merrifield–Simmons index f(G) of a (molecular) graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By we denote the set of unicycle graphs in which the length of its unique cycle is k. In this paper, we investigate the Merrifield–Simmons index f(G) for an unicycle graph G in . Unicycle graphs with the largest or smallest Merrifield–Simmons index are uniquely determined.     

Modification of Wiener Index and its application

A novel topological index based on the Wiener Index is proposed as W* = 1/2 sigma (n)(i,j=1) S(*)(ij), the element S(*)(ij) of the distance matrix is defined either as S(*)(ij) = alpha x square root of I(i)I(j)/R(ij) (atoms i and j are adjacent) or as S(*)(ij) = = alpha x (j-i+1)square root of I(i) x x x x x I(j)/R(ij) (atoms i and j are not adjacent), where I(i) and I(j) represent the electronegativity of vertices i or j, respectively, R(ij)() is the sum of the bond length between the vertices i and j in a molecular graph, and alpha = (Z(i)/Z(j))(0.5), where Z(i) and Z(j) are the atomic numbers of the positive valence atom i and the negative valence atom j, respectively. The properties and the interaction of the vertices in a molecule are taken into account in this definition. That is why the application of the index W to heteroatom-containing and multiple bond organic systems and inorganic systems is possible. Correlation coefficients above 0.97 are achieved in the prediction of the retention index of gas chromatography of the hydrocarbons, the standard formation enthalpy of methyl halides, halogen-silicon, and inorganic compounds containing transition metals.