Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

The proof of a conjecture concerning acyclic molecular graphs with maximal Hosoya index and diameter 4

The Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we solve a conjecture in Ou, J. Math. Chem, DOI: 10.1007/S10910-006-9199-1 concerning acyclic molecular graphs with maximal Hosoya index and diameter 4. Partially supported by NNSFC (No. 10571105, 10671081).     

On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter

The total number of matchings of a graph is defined as its Hosoya index. Conjugated and non-conjugated acyclic graphs that have maximal Hosoya index and short diameter are characterized in this paper, explicit expressions of the Hosoya indices of these extremal graphs are also presented.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Chemical trees minimizing energy and Hosoya index

The energy of a molecular graph is a popular parameter that is defined as the sum of the absolute values of a graph’s eigenvalues. It is well known that the energy is related to the matching polynomial and thus also to the Hosoya index via a certain Coulson integral. It is quite a natural problem to minimize the energy of trees with bounded maximum degree—clearly, the case of maximum degree 4 (so-called chemical trees) is the most important one. We will show that the trees with given maximum degree that minimize the energy are the same that have been shown previously to minimize the Hosoya index and maximize the Merrifield-Simmons index, thus also proving a conjecture due to Fischermann et al. Finally, we show that the minimum energy grows linearly with the size of the trees, with explicitly computable growth constants that only depend on the maximum degree.     

Double hexagonal chains with maximal Hosoya index and minimal Merrifield–Simmons index

It is well known that the two graph invariants, “the Hosoya index” and “the Merrifield–Simmons index” are important ones in structural chemistry. The extremal hexagonal chains with respect to the Hosoya index and Merrifield–Simmons index are determined by Gutman and Zhang (J. Math. Chem., 12 (1993) 197–210, 27 (2000) 319–329 and J. Sys. Sci. Math. Sci., 18 (4) (1998) 460–465). In this paper, we will consider a type of the pericondensed hexagonal system. The double hexagonal chains with maximal Hosoya index and minimal Merrifield Simmons index are determined.     

On the matching polynomials of graphs with small number of cycles of even length

Suppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G^e) of an arbitrary orientation G^e of G and the minors of A(G^e). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Hosoya polynomials of TUC4C8(S) nanotubes

The Hosoya polynomial of a chemical graph G is , where d _G (u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived. This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809).     

Characteristic polynomials of chemical graphs via symmetric function theory

The characteristic polynomial corresponding to the adjacency matrix of a graph is constructed by using the traces of the powers of the adjacency matrix to calculate the coefficients of the characteristic polynomial via Newton's identities connecting the power sum symmetric functions and the elementary symmetric functions of the eigenvalues. It is shown that Frame's method, very recently employed by Balasubramanian, is nothing but symmetric functions and Newton's identities.     

The Merrifield–Simmons Indices and Hosoya Indices of Trees with <Emphasis Type="Italic">k</Emphasis> Pendant Vertices

The Merrifield–Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we characterize the trees with maximal Merrifield–Simmons indices and minimal Hosoya indices, respectively, among the trees with k pendant vertices.     

Operator technique for obtaining the recursion formulas of characteristic and matching polynomials as applied to polyhex graphs

A simple and efficient method, called an operator technique, for obtaining the recurrence relation of a given counting polynomial, e.g., characteristic P_G(x) or matching M_G(x) polynomial, for periodic networks is proposed. By using this technique the recurrence relations of the P_G(x) and M_G(x) polynomials for the linear zigzag-type and kinked polyacene graphs were obtained. For the lower members of these series of graphs, the coefficients of P_G(x) and M_G(x) polynomials are tabulated.     

用于饱和链烃类化合物的结构/性质相关性研究的一种新的拓扑指数

分子的结构决定分子的性质 ,从分子结构提炼出的结构信息进行数值化即得到分子拓扑指数。用拓扑指数来进行定量结构 活性相关性(ＱＳＡＲ)以及定量结构 性质相关性 (ＱＳＰＲ)研究是目前非常活跃的领域。到目前为止 ,已相继提出1 2 0多种拓扑指数[1] 。利用这些拓扑指数来讨论饱和链烃类化合物的性质已经得到了许多有益的结果[2 ] 。本文利用距离矩阵和邻接矩阵 ,提出了一种新的拓扑指数Ｚｈ,对饱和链烃类化合物的性质进行预测和预报 ,结果表明 ,这种拓扑指数具有良好的结构选择性和性质相关性。1　方法以 2 甲基丁烷为例 ,其隐氢图为 :其…     

Bond index: relation to second-order density matrix and charge fluctuations

It is shown that, in the same way as the atomic charge is an invariant built from the first-order density matrix, the closed-shell generalized bond index is an invariant associated with the second-order reduced density matrix. The active charge of an atom (sum of bond indices) is shown to be the sum of all density-density correlation functions between it and the other atoms in the molecule; similarly, the self-charge is the fluctuation of its total charge.On leave of absence from Depto. de Física e Química, UFES, 29000 Vitória, ES, Brasil     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

主族金属元素电子脱出功与拓扑指数~mP的相关性

利用Pauling电负性xpi、原子的价电子数mi和原子的价电子层数ni构建了点价iδ.由点价iδ构建了拓扑指数mP.利用其0阶指数0P与23种金属元素电子脱出功关联,拟合成3个线性回归方程.相关系数为0.9803,0.9870和0.9878,均优于文献值.预测取得了较好的结果.     

A relationship between the retention indices on nematic and isotropic phases and the shape of polycyclic aromatic hydrocarbons

Summary An equation has been derived allowing to predict retention index values on nematic phases from computable parameters and sizes of the molecules of polycyclic aromatic hydrocarbons. Another equation, which has also been derived, allows to determine the shape parameter of a molecule from chromatographic data.     

The study of the relationship between the new topological index A(m) and the gas chromatographic retention indices of hydrocarbons by artificial neural networks

The newly developed topological indices A_m1-A_m3 and the molecular connectivity indices ^mX were applied to multivariate analysis in structure-property correlation studies. The topological indices calculated from the chemical structures of some hydrocarbons were used to represent the molecular structures. The prediction of the retention indices of the hydrocarbons on three different kinds of stationary phase in gas chromatography can be achieved applying artificial neural networks and multiple linear regression models. The results from the artificial neural networks approach were compared with those of multiple linear regression models. It is shown that the predictive ability of artificial neural networks is superior to that of multiple linear regression method under the experimental conditions in this paper. Both the topological indices ²X and A_m1 can improve the predicted results of the retention indices of the hydrocarbons on the stationary phase studied.     

The delocalization index as an electronic aromaticity criterion: application to a series of planar polycyclic aromatic hydrocarbons

This work introduces a new local aromaticity measure, defined as the mean of Bader's electron delocalization index (DI) of para-related carbon atoms in six-membered rings. This new electronic criterion of aromaticity is based on the fact that aromaticity is related to the cyclic delocalized distribution of pi-electrons. We have found that this DI and the harmonic oscillator model of aromaticity (HOMA) index are strongly correlated for a series of six-membered rings in eleven planar polycyclic aromatic hydrocarbons. The correlation between the DI and the nucleus-independent chemical shift (NICS) values is less remarkable, although in general six-membered rings with larger DI values also have more negative NICS indices. We have shown that this index can also be applied, with some modifications, to study of the aromaticity in five-membered rings.