Iranian Journal of Mathematical Chemistry
Volume:10 Issue: 3, Summer 2019

The Bertz indices, derived by counting the number of connecting edges of line graphs of a molecule were used in deriving the QSPR models for the physicochemical properties of alkanes. The inability of these indices to identify the hetero centre in a chemical compound restricted their applications to hydrocarbons only. In the present work, a novel molecular descriptor has been derived from the weighted line graph of the molecular structure and applied in correlating the physicochemical properties of alkane isomers with these descriptors. A weight is tagged at the vertex of the line graph, which consequently modifies the weight of the edge. These descriptors were found to classify the alkane isomers and served well in deriving the QSPR models for various physicochemical properties. The mathematical calculations include the quantitative treatment on the role of substituents (alkyl) in governing the properties under study of the alkane isomers. Further, the use of weighted line graph in the enumeration of the topological index opens up a new vista on application to heteroatomic systems.

In this paper, we compute the Wiener index, first Zagreb index, second Zagreb index, degree distance index and Gutman index of edge corona of two graphs. Also in some cases we derive formulas for Weiner index, Zagreb indices, degree distance and Gutman index in terms of vertices and edges .

The distinguishing number (index) D(G) (D'(G)) of a graph G is the least integer d such that G has an vertex labeling (edge labeling) with d labels that is preserved only by a trivial automorphism. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,... ,Gk by selecting a vertex of G1, a vertex of G2, and identifying these two vertices. Then continue in this manner inductively. We say that G is obtained by point-attaching from G1, ... ,Gk and that Gi's are the primary subgraphs of G.  In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their distinguishing number and distinguishing index.

 Let G = (V,E) be a finite and simple graph with λ1, λ2,...,λn as its eigenvalues.The Estrada index of G is EE(G) =∑ni=1e^{λi} . A spiro compound is a chemical compound that presents a twisted structure of two or more rings, in which 2 or 3 rings are linked together by one common atom. In this paper, we show that the symmetric and stable spiro compounds among all spiro compounds have the minimum Estrada index.

The main aim of the current paper is to propose an upwind local radial basis functions-finite difference (RBF-FD) method for solving compressible Euler equation. The mathematical formulation of chemically reacting, inviscid, unsteady flows with species conservation equations and finite-rate chemistry is studied. The presented technique is based on the developed idea in [58]. For checking the ability of the new procedure, the compressible Euler equation is solved. This equation has been classified in category of system of advection-diffusion equations. The solutions of advection equations have some shock, thus, special numerical methods should be applied for example discontinuous Galerkin and finite volume methods. Moreover, two problems are given that show the acceptable accuracy and efficiency of the proposed scheme.

The topological efficiency index of a connected graph G, denoted by rho (G), is defined as rho(G)=frac{2W(G)}{left|V(G)right|underline w(G)}, where underline w(G)=text { min }left{w_v(G):vin V(G)right} and W(G) is the Wiener index of G. In this paper, we obtain the value of topological efficiency index for some composite graphs such as tensor product, strong product, symmetric difference and disjunction of two connected graphs. Further, we have obtained the topological efficiency index for a double graph of a given graph.