Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Generalized Wheland polynomial for large graphs

We report the generalized Wheland polynomial for acyclic graphs depicting polyenes havingn = 10 carbon atoms. We consider the problem of deriving generalized Wheland polynomials for larger chains by recursion. The recursion Wh(n + l;x) =, Wh(n; x) + (1 –x)Wh(n – 1;x) allows one to find the next larger generalized Wheland polynomial for a chain having an even number of vertices by knowing generalized Wheland polynomials of chains having fewer vertices. The recursion, however, does not allow one to predict the generalized Wheland polynomial for a chain having an odd number of vertices from smaller chains! Here we report a procedure which allows one to derive the generalized Wheland polynomial for a chain having an odd number of vertices. This is achieved by combining the coefficients for rings having the same number of vertices. The generalized Wheland polynomials for odd rings are simply related to the generalized Wheland polynomials for smaller chains and can be derived from the information on smaller chains. This makes it possible to extend the recursion for generalized Wheland polynomials for arbitrarily largen.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x _r be closed points in general position in projective spacePn, then the linear subspaceV ofH ⁰ (⨑ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ⨑ⁿ) formed by those polynomials which are singular at eachx _i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x _r. As such, the “expected” value for the dimension ofV is max(0,h ⁰(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

On the matching polynomials of graphs with small number of cycles of even length

Suppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(G^e) of an arbitrary orientation G^e of G and the minors of A(G^e). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

The use of chebyshev polynomials orthogonal on a finite arbitrary system of points for interpolating changes in nematic-isotropic liquid phase transition temperatures in homologous series

Chebyshev polynomials Ψ _q (x) orthogonal on a finite arbitrary system of points x _i (i = 1−N) are used to interpolate changes in nematic-isotropic liquid phase transition temperatures t _c(x) in homologous series of liquid crystals (x = 1/n, where n is the number of the homologue). The expansion of the t _c(x) function into a series in Ψ _q (x) polynomials was found to be very effective. Already at q ≤ 3, this series describes the known types of the t _c(x) dependences with high accuracy and very small root-mean-square deviations for mesogenic molecules of various chemical structures and dimensions. The dependence of the limiting t _l = t _c(0) value on the form of X-shaped molecules and linear dimensions of N-mers with N rigid aromatic fragments linked with each other by flexible spacer chains was studied.

     

Structural study by X-ray diffraction, magnetic susceptibility and Mössbauer spectroscopy of the Na2Mn2(1 − x)Cd2xFe(PO4)3 (0 ≤ x ≤ 1) solid solution

Na₂Mn_{2(1 − x)}Cd_2xFe(PO₄)₃ (0 ≤ x ≤ 1) phosphates were prepared by solid state reaction and characterized by powder X-ray diffraction, magnetic susceptibility and Mössbauer spectroscopy. The X-ray diffraction patterns indicated the formation of a continuous solid solution which crystallizes in the alluaudite structural type characterized by the general formula X(2)X(1)M(1)M(2)₂(PO₄)₃. The cation distribution, deduced from a structure refinement of the x = 0, 0.5 and 1 compositions, is ordered in the X(2) sites and disordered in the remaining X(1), M(1) and M(2) sites. The magnetic susceptibility study revealed an antiferromagnetic behaviour of the studied compounds. The ⁵⁷Fe Mössbauer spectroscopy confirmed the structural results and proved the exclusive presence of Fe³⁺ ions.     

Homomorphism polynomials of chemical graphs

We say that a graphG ishomomorphic to a graphH if there is a mappingp from the vertices of G onto the vertices ofH such thatp(u) andp() are adjacent inH wheneveru and are adjacent in G. Thehomomorphism polynomial of a graphG is a polynomial in two variables that counts the number of homomorphisms ofG onto the complete graph of each order. This polynomial can be computed recursively in an analog to the chromatic polynomial. In this paper, we present some results regarding the homomorphism polynomials of the graphs of chemical compounds — in particular, alkane isomers. The coefficients of the homomorphism polynomial can be used to predict the rankings of compounds with respect to several chemical properties. Our results seem to refine those obtained by Randi et al. from path lengths.     

A möbius inversion of the ulam subgraphs conjecture

The generalized Eichinger matrices are defined asE = _j ⁿ ₁( _j S ^T S)^–1, where _j M denotes the matrixM withj th row and column deleted.S is the incidence matrix andM ^T is the transposed matrix. The conjectureS ^T SE = S _K ^TS _K , where SK is the incidence matrix of the complete graph, is proven for trees, simple cycles and complete graphs. The consequence of the conjecture isS _G ^T S _G (E _G -I) = S _G ^TS _G , whereG is the complementary graph ofG. It leads to graphs with imaginary arcs as the complements of graphs with multiple arcs.     

M5(PO4)3F (M=Ca, Sr, Ba)中Sm3+的电荷迁移态及Sm3+和Eu3+的电荷迁移能量关系

采用高温固相反应合成了M_5-2xSm_xNa_x(PO₄)₃F(M=Ca,Sr,Ba)荧光体,研究了其在真空紫外-可见光范围的发光特性。发现在Ca₅(PO₄)₃F中Sm³⁺的电荷迁移带约在191 nm,在Sr₅(PO₄)₃F中约在199 nm,而在Ba₅(PO₄)₃F中约在204 nm,随着被取代碱土离子半径的增大电荷迁移能量逐渐减小。比较了M₅(PO₄)₃F (M=Ca,Sr,Ba)中Sm³⁺和Eu³⁺电荷迁移能量的关系。     

Ordering chemical unicyclic graphs by Wiener polarity index

Suppose G is a chemical graph with vertex set V(G). Define D(G) = {{u, v} ⊆ V (G) | d _G (u, v) = 3}, where d _G (u, v) denotes the length of the shortest path between u and v. The Wiener polarity index of G, W _p (G), is defined as the size of D(G). In this article, an ordering of chemical unicyclic graphs of order n with respect to the Wiener polarity index is given.     

Nitrous Oxide as Carrier Gas in Capillary Gas-Liquid Chromatography

On use of nitrous oxide as carrier gas the retention factors of the chromatographed compounds decrease linearly with increasing average column pressure. Other retention characteristics (relative retention, retention index) change linearly. This effect was demonstrated by using a capillary column coated with nonpolar polydimethylsiloxane phase SE-30. As shown for capillary GLC the linear correlation is valid for the same column:k_i(G₁,P₁) = A k_i(G₂, P₂) + B, where k_i(G₁, P₁) and k_i(G₂,P₂) are the retention factors of compound i at average column pressures P₁ and P₂ when using carrier gases G₁ and G₂, respectively; A and B are coefficients.     

Corrigendum on ‘investigation on molecular interaction studies of binary mixture of Dehpa and Petrofin at 298.15 K’

Expressions of Redlich–Kister (RK) excess function using polynomials on difference of molar composition are used. Calculated values of excess properties from experimental values are well presented in some curves as a function of mole fraction (x₂) of the second pure component of the studied binary liquid mixture. Nevertheless, the authors presented values of adjustable parameters, not for the polynomial of the popular RK equation (i.e. vs. x₂–x₁), but for globally the excess property Y^E as an overall three degree polynomial on (x₂) including implicitly the molar fraction product (x₁·x₂) in their nonlinear regression, which can then induces some readers to probable confusion.     

Imminant polynomials of graphs

Summary The imminant polynomials of the adjacency matrices of graphs are defined. The imminant polynomials of several graphs [linear graphs (L _n ), cyclic graphs (C _n ) and complete graphs (K _n )] are obtained. It is shown that the characteristic polynomials and permanent polynomials become special cases of imminant polynomials. The connection between the Schur-functions and imminant polynomials is outlined.Cammile & Henry Dreyfus Teacher-scholar