Iranian Journal of Mathematical Chemistry
Volume:4 Issue: 2, Summer-Autumn 2013

The purpose of this paper is to analyze in detail a special nonlinear partial differential equation (nPDE) of the second order which is important in physical, chemical and technical applications. The present nPDE describes nonlinear diffusion and is of interest in several parts of physics, chemistry and engineering problems alike. Since nature is not linear intrinsically the nonlinear case is therefore the general. We determine the classical Lie point symmetries including algebraic properties whereas similarity solutions are given as well as nonlinear transformations could derived. In addition, we discuss the nonclassical case which seems to be not solvable. Moreover we show how one can deduce approximate symmetries modeling the nonlinear part and we deduce new generalized symmetries of lower symmetry. The analysis allows one to deduce wider classes of solutions either of practical and theoretical usage in different domains of science and engineering.

We can reach by DNA microarray gene expression to such wealth of information with thousands of variables (genes). Analysis of this information can show genetic reasons of disease and tumor differences. In this study we try to reduce high-dimensional data by statistical method to select valuable genes with high impact as biomarkers and then classify ovarian tumor based on gene expression data of two patient groups. One group treated by standard therapies and survived, while another group didn’t be cure and die after some times. In the first step we used weighted voting algorithm (WVA) for selecting impressive genes to reduce dimension, therefore eliminate noisy data and make analysis easier and then partial least square – discriminante analysis (PLS-DA) and support vector machine (SVM) methods have been applied for classification of diminished data. Results show that classification by PLS-DA can distinguish two groups somewhat but SVM is more efficient and sufficient classification method.

Let G be a simple connected graph. The generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.

In the present work, DFT calculations are employed to obtain the optimized structures of 4- acyl pyrazolone tautomers (19 tautomers) using B3LYP/6-311++G** calculations. In addition, molecular parameters, IR frequencies and relative energies are extracted for all tautomers. The existence of aromatic ring, keto tautomer (versus enol tautomer), N-H bond (versus C-H bond) and C=N double bond (versus N=N double bond) are stabilizing factors in relative stabilities of tautomers. Calculation of vibrational frequencies showed that, in accordance with reported values, intramolecular hydrogen bond (existed in some tautomers) decreased the value of OH frequency. The solvent effects on relative stabilities of tautomers are calculated. The relative stabilities of all the tautomers in acetone, tetrahydrofurane and chloroform (in all solvents, except water) were relatively the same as those in the gas phase. In addition, a nearly good relationship is found between dipole moments of tautomers and their 7Gsolv in chloroform. This relation shows that by increasing the dipole moment, the absolute amount of 7Gsolv in chloroform increases.

In this paper, we introduce the new class of implicit L-stable generalized hybrid methods for the numerical solution of first order initial value problems. We generalize the hybrid methods with utilize ynv directly in the right hand side of classical hybrid methods. The numerical experimentation showed that our method is considerably more efficient compared to well known methods used for the numerical solution of stiff first order initial value problems.

Let G be a simple connected graph. The first and second Zagreb indices have been introduced as  vV(G) (v)2 M1(G) degG and M2(G)  uvE(G)degG(u)degG(v), respectively, where degG v(degG u) is the degree of vertex v (u). In this paper, we define a new distance-based named HyperZagreb as e uv E(G). (v))2 HM(G)     (degG(u)  degG In this paper, the HyperZagreb index of the Cartesian product, composition, join and disjunction of graphs are computed.

In this paper, we first collect the earlier results about some graph operations and then we present applications of these results in working with chemical graphs.

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x, where, d(u, v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific graphs.

The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices of G. The eccentricity of a vertex v in G is the distance to a vertex farthest from v. In this paper we obtain the Wiener index of a graph in terms of eccentricities. Further we extend these results to the self-centered graphs.

Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 {,} () (() ())[(,)] RDD(G) = u v V G d u  d v d u v, where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

Let G be a simple graph with vertex set V(G) {v1,v2, ...vn}. For every vertex i v, () i  v represents the degree of vertex i v. The h-th order of Randić index, h R is defined as the sum of terms 1 2 1 1 (), ()... () i i ih  v  v  v  over all paths of length h contained (as sub graphs) in G. In this paper, some bounds for higher Randić index and a method for computing the higher Randic index of a simple graph is presented. Also, the higher Randić index of coronene/circumcoronene is computed.

The atom-bond connectivity index of graph is a topological index proposed by Estrada et al. as ABC (G)  uvE (G) (du dv  2) / dudv, where the summation goes over all edges of G, du and dv are the degrees of the terminal vertices u and v of edge uv. In the present paper, some upper bounds for the second type of atom-bond connectivity index are computed.