Sum of characteristic polynomial coefficients of cycloparaphenylene graphs as topological index

ABSTRACT

Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.     

Analytical eigenspectra of alternant edge-weighted graphs of linear chains and cycles: some applications

Analytical eigenspectra for the graphs of linear chains and cycles with alternant edge weights has been derived with the use of two independent methods, namely, the characteristic polynomial and the graph squaring. In the former method the rotational symmetry and the trigonometric identity have been exploited. These methods along with the expressions of eigenspectra so obtained have been found to be very useful in expressing analytical eigensolutions of some important as well as novel benzenoids, for example, linear p-methylene poly(p-phenylene), cylindrical poly(p-phenylene), zigzag edge graphene, carbon nanotube and carbon nanotori. Some of these eigensolutions have been analysed in exploring some consequences thereof.     

Matching polynomial coefficients and the Hosoya indices of poly(p-phenylene) graphs

A general approach to determine the matching polynomial (MP) of a graph with two parts connected by an edge is presented in matrix product that is ultimately used in deducing recursion formulas for obtaining the MP coefficients of linear and cylindrical poly(p-phenylene) (PPP) graphs. The Hosoya indices of linear and cylindrical PPPs are derived in terms of that of the two immediately preceding graphs as well as in analytical forms with the use of transfer matrices. Ambient condition density and bulk modulus of linear PPPs with 2–6 phenyl rings have been found to correlate well with the logarithm of their Hosoya indices. Excellent correlations of diameters with the logarithm of Hosoya indices and strain energies with the inverse of the logarithm of Hosoya indices for cylindrical PPPr with r (= 6–16, 18, 20) phenyl rings are obtained. The linear relation between the logarithm of Hosoya indices and diameter and the inverse relation between diameter and strain energy corroborate the fact.     

A Pascal's triangle-like approach for the determination of characteristic polynomial coefficients of reciprocal graphs

Characteristic polynomial coefficients of three classes of graph, namely L_n + n(p), C_n + n(p) and K _1,n-1 + n(p), which are known to have reciprocal pairs of eigenvalues, have been shown to be generated by simple manipulation of the Pascal's triangle.     

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACT

Three classes of reciprocal graphs, viz. monocycle (G^C_n), linear chain (G^L_n) and star (G^K_n) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (G^C_n) and linear chain (G^L_n) are obtained by putting a pendant vertex to each vertex of simple monocycle (C_n) and simple linear chain (L_n), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n ? 1) peripheral vertices of the star graph (K_{1, (n?1)}). An n-fold rotational axis of symmetry for G^C_n and (n ? 1)-fold rotational axis of symmetry for G^K_n have been exploited for obtaining their respective condensed graphs. The condensed graph for G^L_n has been generated from that of G^C_n incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.     

The Tutte polynomial of an infinite family of outerplanar,small-world and self-similar graphs

In this paper we recursively describe the Tutte polynomial of an infinite family of outerplanar, small-world and self-similar graphs. In particular, we study the Abelian Sandpile Model on these graphs and obtain the generating function of the recurrent configurations. Further, we give some exact analytical expression for the Tutte polynomial at several special points     

Use of symmetry plane fragmentation and graph squaring techniques to express the eigenspectra for some vertex-weighted graphs of linear chains and cycles in analytical form

Three types of graphs of linear chains, viz. linear chains with unit increment or decrement in weight on one terminal vertex, linear chains with unit increment or decrement in weight on both the terminal vertices and linear chains with unit increment in weight on one terminal vertex and decrement in that on the other terminal vertex, have been considered. The symmetry plane fragmentation and graph squaring techniques have been exploited to express the eigenspectra of such graphs of linear chains in analytical form, and have subsequently been used to express the eigenspectra of graphs of linear chains and cycles with alternant vertex weights. The derived expressions for the eigenspectra have been used to obtain the eigenspectra of linear polyacenes, methylene-substituted linear polyacenes and cylindrical polyacene strips in analytical form.     

多模玻色二次多项式型系统的特性函数和准概率分布函数

运用多模玻色系统广义线性量子变换的普遍理论，对多模玻色二次多项式型系统进行了研究.给出了多模玻色二次多项式型系统的正规、反正规、Wigner特性函数、P表示和Q表示.举例讨论了该系统的压缩性质. 关键词： 多模玻色二次多项式型系统 特性函数 准概率分布函数     

Wigner function and tomogram of the Hermite polynomial state

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

Efficiency of star-like graphs and the Atlanta subway network

The distance d(i,j)$d(i,j)$ between any two vertices i$i$ and j$j$ in a graph is the number of edges in a shortest path between i$i$ and j$j$. If there is no path connecting i$i$ and j$j$, then d(i,j)=∞$d(i,j)=\infty$. In 2001, Latora and Marchiori introduced the measure of efficiency between vertices in a graph (Latora and Marchiori, 2001) [1]. The efficiency between two vertices i$i$ and j$j$ is defined to be _∈i,j=j${\in }_{i,j}=j$. In this paper, we investigate the efficiency of star-like networks, and show that networks of this type have a high level of efficiency. We apply these ideas to an analysis of the Metropolitan Atlanta Rapid Transit Authority (MARTA) Subway system, and show this network is 82% as efficient as a network where there is a direct line between every pair of stations.     

Convergent polynomial expansion,scaling of differential cross-section and computation of scaling functions for elastic diffraction scattering processes

Existing data on the differential cross-section ratio at high energies for pp, p, π^±p andK ^±p scatering have been fitted by the proposed convergent polynomial expansion to determine the unknown coefficients in the scaling function. It is found that the data are very well represented within and somewhat outside the peak regions by only four or five terms in the proposed series in terms of Laguerre polynomials.     

Recurrent random walks and the absence of continuous symmetry breaking on graphs

We consider geometrically disordered systems with a continuous symmetry groupG, where the internal degrees of freedom are attached to the vertices of a graph. We show that equilibrium states remainG-invariant at any temperatureT>0 if a random walk on the graph is recurrent. This generalizes a previous result obtained by Cassi.     

Modified 2CLJDQP model and the second virial coefficients for linear molecules

A modified form of 2CLJDQP potential model is proposed to calculate the second virial coefficients of two-center Lennard-Jones molecules. In the presented potential model, the potential parameters σ and ε are considered as the temperature-dependent parameters in the form of hyperbolical temperature function based on the theory of temperaturedependent potential parameters. With this modified model, the second virial coefficients of some homonuclear molecules（such as O2, Cl2, CH3CH3, and CF3CF3） and heteronuclear molecules（such as CO, NO, CH3 F, CH3 Cl, CH3CF3,CH3CHF2, and CF3CH2F） are calculated. Then the Lorentz–Berthelot mixing rule is modified with a temperaturedependent expression, and the second virial coefficients of the heteronuclear molecules（such as CH3 F, CH3 Cl, and CH3CF3） are calculated. Moreover, CO2 and N2O are also studied with the modified 3CLJDQP model. The calculated results from the modified 2CLJDQP model accord better with the experimental data than those from the original model.It is shown that the presented model improves the positive deviation in low temperature range and negative deviation in high temperature range. So the modified 2CLJDQP potential model with the temperature-dependent parameters can be employed satisfactorily in large temperature range.     

Cluster expansion and generalized transfer matrices for the statistical mechanics of linear chains

A cluster expansion of the statistical mechanical density operator for a general linear chain model with nearest-neighbor interactions is made. This expansion is then shown to lead to an expansion of a generalized transfer matrix, whose maximum eigenvalue is the per-site partition function. A number of computational features, as well as some illustrative examples, of this approach are described.Research supported in part by the Robert A. Welch Foundation, Houston, Texas.     

改进的测二极管伏安特性的电路

指出了某普通物理实验教材中所给出的伏安法测二极管特性的电路所存在的总理２,并 合理的电路。     

An Accurate and Efficient Algorithm for the Computation of the Characteristic Polynomial of a General Square Matrix

An algorithm is presented for the efficient and accurate computation of the coefficients of the characteristic polynomial of a general square matrix. The algorithm is especially suited for the evaluation of canonical traces in determinant quantum Monte-Carlo methods.     

Convergent polynomial expansion,energy dependence of slope parameters,scaling hypothesis and predictions forπ ± p andK + p scattering

A phenomenological representation for differential cross-section recently proposed using Mandelstam analyticity and convergent polynomial expansion (CPE) which has been found to be successful in describing scaling of the differential cross-section-ratio data for several elastic diffractive and inelastic nondiffractive processes is used to analyse the energy dependence of the slope-parameter data at high energies, extrapolate the slope parameter and predict the differential cross-section ratio as a function of |t| at higher energies forπ ^± pndK ⁺ p scattering. Following the method of Hansen and Krisch it is found that, in spite of the existence of rather widely varying data points for nearbys values, a more systematic trend in the energy dependence of the slope parameter emerges when a statistical average of the existing high-energy data is used. Extrapolating the fits to the average data ontos → ∞ provides strong evidence in favour of a model-independent result that asymptotically theπ ^± p slopes may be equal. There is also a strong indication to the effect that each of these two slopes may be equal to theK ⁺ p slope fors → ∞. Using the scaling curves generated by the existing data on differential cross-section ratio and extrapolated values of the slope parameter, the differential cross-section ratio for each of the three processes is predicted as a function of |t| for higher energies.     

Derivation of the linear and nonlinear non-Markov fluctuation-dissipation relations of the first kind for a dynamical model

A dynamical system consisting of a subsystem having the variablesz=(q,p) and of another dynamical system (thermostat) is considered in the nonquantum case. Using a dynamical equation, it is shown that the linear and quadratic non-Markov fluctuation-dissipation relations (FDRs) of the first kind are valid in the first nonvanishing approximation in interaction constants. Applying these FDRs, one can determine the statistical properties of the fluctuations when the form of the nonlinear phenomenological equation is known. The non-Markov FDRs of the first kind are the direct generalization (to the inertial case) of the Markov FDRs that are the consequence of detailed balance.     

“西林”椭圆曲线的特性分析及其应用

本文对“西林”椭圆曲线作了定最描述,并就其特性进行了分析。作为数字例子,文中最后给出两个实际应用的设计结果。