- Volume:9 Issue:2, 2018
- تاریخ انتشار: 1397/03/27
- تعداد عناوین: 7
An algebraic calculation method for describing time-dependent processes in electrochemistry - Expansion of existing proceduresPages 77-100In this paper an alternative model allowing the extension of the Debye-Hückel Theory (DHT) considering time dependence explicitly is presented. From the Electro-Quasistatic approach (EQS) introduced in earlier studies time dependent potentials are suitable to describe several phenomena especially conducting media as well as the behaviour of charged particles (ions) in electrolytes. This leads to a reformulation of the meaning of the nonlinear Poisson-Boltzmann Equation (PBE). If a concentration and/or flux gradient of particles is considered the original structure of the PBE will be modified leading to a nonlinear partial differential equation (nPDE) of the third order.
It is shown how one can derive classes of solutions for the potential function analytically by application of pure algebraic steps. The benefit of the mathematical tools used here is the fact that closed-form solutions can be calculated and thus, numerical methods are not necessary.
The important outcome of the present study is twofold meaningful: (i) The model equation allows the description of time dependent problems in the theory of ions, and (ii) the mathematical procedure can be used to derive classes of solutions of arbitrary nPDEs, especially those of higher order.Keywords: Nonlinear partial differential equations (nPDEs), nonlinear ordinary differential equations (nODEs), Debye, Hückel Theory (DHT), Poisson, Boltzmann Equation (PBE)
Pages 101-111For a graph G, the irregularity and total irregularity of G are defined as irr(G)=∑_(uv∈E(G))〖|d_G (u)-d_G (v)|〗 and irr_t (G)=1/2 ∑_(u,v∈V(G))〖|d_G (u)-d_G (v)|〗, respectively, where d_G (u) is the degree of vertex u. In this paper, we characterize all ýconnected Eulerian graphs with the second minimum irregularity, the second and third minimum total irregularity value, respectively.Keywords: Eulerian graphs, irregularity, total irregularity, vertex degree
Pages 113-120Using an identity for effective resistances, we find a relationship between the arithmetic-geometric index and the global ciclicity index. Also, with the help of majorization, we find tight upper and lower bounds for the arithmetic-geometric index.Keywords: arithmetic, geometric index, global cyclicity index, majorization
Novel Atom-Type-Based Topological Descriptors for Simultaneous Prediction of Gas Chromatographic Retention Indices of Saturated Alcohols on Different Stationary PhasesPages 121-135In this work, novel atom-type-based topological indices, named AT indices, were presented as descriptors to encode structural information of a molecule at the atomic level. The descriptors were successfully used for simultaneous quantitative structure-retention relationship (QSRR) modeling of saturated alcohols on different stationary phases (SE-30, OV-3, OV-7, OV-11, OV-17 and OV-25). At first, multiple linear regression models for Kovats retention index (RI) of alcohols on each stationary phase were separately developed using AT and Randics first-order molecular connectivity (1χ) indices. Adjusted correlation coefficient (R2adj) and standard error (SE) for the models were in the range of 0.994-0.999 and 4.40-8.90, respectively. Statistical validity of the models were verified by leave-one-out cross validation (R2cv > 0.99). In the next step, whole RI values on the stationary phases were combined to generate a new data set. Then, a unified model, added McReynolds polarity term as a descriptor, was developed for the new data set and the results were satisfactory (R2adj=0.995 and SE=8.55). External validation of the model resulted in the average values of 8.29 and 8.69 for standard errors of calibration and prediction, respectively. The topological indices well covered the molecular properties known to be relevant for retention indices of the model compounds.Keywords: Quantitative structure–retention relationship, Atom, type, based topological indices, Saturated alcohols, Modeling
Pages 137-147The Laplacian-energy-like of a simple connected graph G is defined as
Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the number of triangles in graph.Keywords: Laplacian spectrum, Laplacian, energy, like invariant, Cauchy, Schwarz inequality, Lagrange identity, spectral radius
Pages 149-156The aim of this paper is to compute some bounds of forgotten index and then we present spectral properties of this index. In continuing, we define a new version of energy namely ISI energy corresponded to the ISI index and then we determine some bounds for it.Keywords: Zagreb indices, forgotten index, ISI index, energy of graph
Pages 157-165Betweenness centrality is a distance-based invariant of graphs. In this paper, we use lexicographic product to compute betweenness centrality of some important classes of graphs. Finally, we pose some open problems related to this topic.Keywords: Betweenness centrality, lexicographic product tensor product, strong product