On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.     

Walking backward: walk counts of negative order

A closed formula is derived for walk counts of negative order k in a graph or molecule, as defined recently by Lukovits and Trinajsti?. Some unexpected observations made by these authors easily follow from this formula. Gratifyingly, the formula is very similar to the one obtained earlier for usual walk counts. Moreover, while for walk counts of k --> + infinity the numerically largest eigenvalue of the adjacency matrix plays an important part, for walk counts of k --> - infinity the numerically smallest eigenvalue plays a corresponding part.     

External factor variable connectivity index

A new variable index, external factor variable connectivity index (EFVCI), is proposed, in which the atomic attribute is divided into two parts. The innate part is denoted as outer-shell electrons and external part or perturbation by other atoms is represented as summation, multiplied by a variable x, of squared reciprocal matrix of i row (corresponds to atom A(i)). The division of atomic attribute in EFVCI is interpreted by using topological structure. In the correlation of boiling point of 149 acyclic alkanes, the optimal values will approach to a constant at -0.29 by using the zero to higher order indices of the same series. The new index, with high regression quality (R = 0.9986, s = 2.26, and F = 7088.4), is compared favorably with variable connectivity index and molecular connectivity index.     

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.     

单硫醚气相色谱保留指数拓扑化学研究

在分子拓扑化学理论的基础上,根据分子中原子的特性,用分子中原子的平衡电负性对分子图进行着色,在距离矩阵的基础上结合分子中各原子的支化度构建一组新的拓扑指数NPm(m=1,2,3),利用多元线性回归技术将单硫醚在4种极性固定相的气相色谱保留指数与NPm(m=1,2,3)建立相应的定量结构-保留相关关系模型(QSRR),并用这种模型对单硫醚的气相色谱保留指数进行预测,结果表明,预测结果和实验值吻合较好。     

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。     

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。     

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.     

Scaling and the Smoluchowski equations

The Smoluchowski equations, which describe coalescence growth, take into account combination reactions between a j-mer and a k-mer to form a (j+k)-mer, but not breakup of larger clusters to smaller ones. All combination reactions are assumed to be second order, with rate constants K(jk). The K(jk) are said to scale if K(lambda j,gamma k) = lambda(mu)gamma(nu)K(jk) for j < or = k. It can then be shown that, for large k, the number density or population of k-mers is given by Ak(a)e(-bk), where A is a normalization constant (a function of a, b, and time), a = -(mu+nu), and b(mu+nu-1) depends linearly on time. We prove this in a simple, transparent manner. We also discuss the origin of odd-even population oscillations for small k. A common scaling arises from the ballistic model, which assumes that the velocity of a k-mer is proportional to 1/square root of m(k) (Maxwell distribution), i.e., thermal equilibrium. This does not hold for the nascent distribution of clusters produced from monomers by reactive collisions. By direct calculation, invoking conservation of momentum in collisions, we show that, for this distribution, velocities are proportional to m(k)(-0.577). This leads to mu+nu = 0.090, intermediate between the ballistic (0.167) and diffusive (0.000) results. These results are discussed in light of the existence of systems in the experimental literature which apparently correspond to very negative values of mu+nu.     

Exchange interaction of 5,5'-(m- and p-phenylene)bis(10-phenyl-5,10-dihydrophenazine) dications and related analogues

[structures: see text] 5,5'-(m-Phenylene)bis[(10-aryl-5,10-dihydrophenazine) dications, 3a2+ and 3b2+, and their p-analogues 4a2+ and 4b2+, were prepared, and their exchange interaction was investigated. The EPR spectra of these dications at 123 K in a butyronitrile matrix showed the population of a triplet state. The temperature dependence of the EPR signal intensity (absolute value(delta m(s)) = 2) showed that these dications had singlet ground states with deltaE(ST)/k(B) = -27 to -21 K for the m-isomer 3(2+) and with deltaE(ST)/k(B) = -10 to -8 K for the p-isomer 4(2+). Theoretical calculation of the exchange interaction J for these dications at the orthogonal torsion angle geometries was carried out for 3a2+ and 4a2+ and for (m- and p-phenylene)bisphenothiazine dications 1(2+) and 2(2+) using the broken-symmetry approach for the singlet states. A good correlation was observed between the calculated J and a MO-energy term in the triplet state, deltaE(TMO) = absolute value(HOMO(alpha) - (HOMO - 1)(alpha)). The calculated J values were negative in the order of 10 K for the m-dications (J/k(B) = -14.7 K for 1(2+), -11.5 K for 3a(2+)), but much smaller negative values were found for the p-isomers (J/k(B) = -0.9 K for 2(2+), -0.8 K for 4a2+). The smaller absolute value(J) values for the p-dications are qualitatively consistent with the experimental deltaE(ST) (2J) values.     

一种新的结构参数用于卤化物的QSPR研究

基于邻接矩阵和原子特征值（ｔｉ）建立一种新的结构参数（＾１Ｑ），它对无机分子具有良好的区分能力，并且计算简单，＾１Ｑ用于ＡＸｋ（ｋ＝１，２，３，４）型卤化物的标准生成焓、晶格能、反应截面等物理化学性质的相关性研究，获得了优于文献方法的结果。     

Computation of the pseudorotation matrix to satisfy the Eckart axis conditions

A general solution for satisfying the Eckart axis conditions [C. Eckart, Phys. Rev. 47, 552 (1935)] is presented. The goal is to find such a pseudorotation matrix T that the vector product between the reference molecular conformation R and another transformed conformation r' is zero [ summation operator(a)m(a) r(a) 'xRa=0; r(a) '=Tr(a)]. Our solution avoids the limitations of the earlier one [H. M. Pickett and H. L. Strauss, J. Am. Chem. Soc. 92, 7281 (1970)], which fails when one of the involved intermediate matrices is singular. We also discuss how to choose among the always nonunique pseudorotation matrices T the one that represents a true rotation for situations when an alignment of the two conformations is desired.     

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.     

Novel atomic-level-based AI topological descriptors: application to QSPR/QSAR modeling

Novel atomic level AI topological indexes based on the adjacency matrix and distance matrix of a graph is used to code the structural environment of each atomic type in a molecule. These AI indexes, along with Xu index, are successfully extended to compounds with heteroatoms in terms of novel vertex degree v(m), which is derived from the valence connectivity delta(v) of Kier-Hall to resolve the differentiation of heteroatoms in molecular graphs. The multiple linear regression (MLR) is used to develop the structure-property/activity models based on the modified Xu and AI indices. The efficiency of these indices is verified by high quality QSPR/QSAR models obtained for several representative physical properties and biological activities of several data sets of alcohols with a wide range of non-hydrogen atoms. The results indicate that the physical properties studied are dominated by molecular size, but other atomic types or groups have small influences dependent on the studied properties. Among all atomic types, -OH groups seem to be most important due to hydrogen-bonding interactions. On the contrary, -OH groups play a dominant role in biological activities studied, although molecular size is also an important factor. These results indicate that both Xu and AI indices are useful model parameters for QSPR/QSAR analysis of complex compounds.     

Modeling boiling points of cycloalkanes by means of iterated line graph sequences

A class of models for predicting boiling points of cycloalkanes is put forward, based on iterated line graphs L(i), i = 1, 2,., of the molecular graph G = L(0). Let m(i) be the number of edges of L(i), i = 0, 1, 2,. The models analyzed are of the form a(0)m(i)()(0) + a(1)m(i)(1) + a(2)m(i)(2) +. + a(k)m(ik) + b. Our optimal QSPR formulas contain m(0), m(1), m(2), m(3), and/or m(4) but never m(5) and m(6). Their precision is as good as or better than the approximations recently reported by Rücker and Rücker (J. Chem. Inf. Comput. Sci. 1999, 39, 788-802).     

Estimation of analytical values from sub-detection limit measurements for water quality parameters

A method is described for estimating analytical values for water quality parameters from sub-detection limit measurements. The method, which is referred to as the error approximation (EA) procedure, relies on quality control analytical procedures and on the assumption that the bulk of the analytical error associated with measurements at or near the detection limit exists within k = -3 to +3 standard deviations for normally (or approximately normally) distributed errors. The EA procedure also assumes that the analytical errors are equally distributed on each side of half the detection limit and that the sub-detection limit value lies between zero and the detection limit.     

Nonbonding orbitals in fullerenes: nuts and cores in singular polyhedral graphs

A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding pi orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (-2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10 190 782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62 043 of the 398 383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies: it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.     

用拓扑指数＆mT预测化合物在热导上的相对物质的量校正因子

在邻接矩阵的基础上,建立一种新的拓扑指数mT,mY=∑(δi·δj·δk…)0.5,其中0T=∑(δi)0.5,1Y=∑(δi·δj)0.5,并计算了10个系列142个分子的0T,1T值.发现mT与这些化合物的气相色谱相对物质的量校正因子有很好的相关性.相关系数均大于0.96,拓扑指数mT能较好地反映化合物的结构特征.