Iranian Journal of Mathematical Chemistry
Volume:8 Issue: 2, Spring 2017

In this paper, a novel topological index, named M-index, is introduced based on expanded form of the Wiener matrix. For constructing this index the atomic characteristics and the interaction of the vertices in a molecule are taken into account. The usefulness of the M-index is demonstrated by several QSPR/QSAR models for different physico-chemical properties and biological activities of a large number of diversified compounds. Moreover, the applicability of the proposed index has been checked among isomeric compounds. In each case the stability of the obtained model is confirmed by the cross validation test. The results of present study indicate that the M-index provides a promising route for developing highly correlated QSPR/QSAR models. On the other hand, the M-index is easy to generate and the developed QSPR/QSAR models based on this index are linearly correlated. This is an interesting feature of the M-index when compared with quantum chemical descriptors which require vast computational cost and exhibit limitations for large sized molecules.

A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

In the present paper, we develop a modified pseudospectral scheme for solving an optimal control problem which is governed by a switched dynamical system. Many real-world processes such as chemical processes, automotive systems and manufacturing processes can be modeled as such systems. For this purpose, we replace the problem with an alternative optimal control problem in which the switching times appear as unknown parameters. Using the Legendre-Gauss-Lobatto quadrature and the corresponding differentiation matrix, the alternative problem is discretized to a nonlinear programming problem. At last, we examine three examples in order to illustrate the efficiency of the proposed method.

Let ýG=(V,E) ýbe aý ýsimple ýconnected ýgraph ýwith ývertex ýset ýVýýý ýand ýedge ýset ýýýE. ýThe Szeged index ýof ýýG is defined by ý where ý respectively ý ý is the number of vertices of ýG ýcloser to ýuý (ýýrespectively v)ý ýýthan ýýýv (ýýrespectively uý).ý
ýýIf ýýýýS ýis aý ýset ýof ýsizeý ý ý ýlet ýýV ýbe ýthe ýset ýof ýall ýsubsets ýof ýýS ýof ýsize ý3. ýThen ýwe ýdefine ýthree ýýtypes ýof ýintersection ýgraphs ýwith ývertex ýset V. These graphs are denoted by ýý ýý and we will find their ýSzeged ýindices.ý

The \emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=\sum {u,v\in V(G)}\frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the \emph{Steiner distance} $d_G(S)$ of the vertices of $S$ is the minimum size of a connected subgraph whose vertex set contains $S$. Recently, Furtula, Gutman, and Katani'{c} introduced the concept of Steiner Harary index and gave its chemical applications. The \emph{$k$-center Steiner Harary index} $SH_k(G)$ of $G$ is defined by $SH_k(G)=\sum_{S\subseteq V(G),|S|=k}\frac{1}{d_G(S)}$. In this paper, we get the sharp upper and lower bounds for $SH_k(G)_k(\overline{G})$ and $SH_k(G)\cdot SH_k(\overline{G})$, valid for any connected graph $G$ whose complement $\overline {G}$ is also connected.

This study presents some mathematical methods for estimating the critical properties of 40 different types of alkanes and their derivatives including critical temperature, critical pressure and critical volume. This algorithm used QSPR modeling based on graph theory, several structural indices, and geometric descriptors of chemical compounds. Multiple linear regression was used to estimate the correlation between these critical properties and molecular descriptors using proper coefficients.
To achieve this aim, the most appropriate molecular descriptors were chosen from among 11 structural and geometric descriptors in order to determine the critical properties of the intended molecules. The results showed that among all the proposed models to predict critical temperature, pressure and volume, a model including the combination of such descriptors as HyperWiener, Platt, MinZL is the most appropriate one.

In this paper, Kekule structures of benzenoid chains are considered. It has been shown that the coefficients of a B_n (x) Morgan-Voyce polynomial equal to the number of k-matchings (m(G,k)) of a path graph which has N=2n points. Furtermore, two relations are obtained between regularly zig-zag nonbranched catacondensed benzenid chains and Morgan-Voyce polynomials and between regularly zig-zag nonbranched catacondensed benzenid chains and their corresponding caterpillar trees.