Estimation of partial linear error-in-variables models for ρ<Superscript>−</Superscript>-mixing dependence data

Consider the partly linear regression model Y = xβ + g(t) + e where the explanatory x is erroneously measured, and both t and the response Y are measured exactly, the random error e is ρ⁻-mixing. Let be a surrogate variable observed instead of the true x in the primary survey data. Assume that in addition to the primary data set containing N observations of , which is ρ⁻-mixing data sets, an independent validation data containing n observations of is available. The exact observations on x may be obtained by some expensive or diffcult procedures for only a small subset of subjects enrolled in the study. In this paper, inspired by Berberan-Santos et al. [J. Math. Chem. 37 (2005)101], a semiparametric method with the primary data is employed to obtain the estimators of β and g(·) based on the least squares criterion with the help of validata. The proposed estimators are proved to be strongly consistent.      

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

Unicycle Graphs with Maximum Generalized Topological Indices

Let G be an unicycle graph and d _v the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph , , where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized. This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan Province(02C210)(04B047).     

Characterizations for Some Types of DNA Graphs

Vertex induced subgraphs of directed de Bruijn graphs with labels of fixed length k and over α letter alphabet are (α,k)-labelled. DNA graphs are (4,k)-labelled graphs. Pendavingh et al. proved that it is NP-hard to determine the smallest value α _k (D) for which a directed graph D can be (α _k (D),k)-labelled for any fixed . In this paper, we obtain the following formulas: and for cycle C _n and path P _n . Accordingly, we show that both cycles and paths are DNA graphs. Next we prove that rooted trees and self-adjoint digraphs admit a (Δ,k)-labelling for some positive integer k and they are DNA graphs if and only if Δ ≤ 4, where Δ is the maximum number in all out-degrees and in-degrees of such digraphs.     

On zeroth-order general Randić index of conjugated unicyclic graphs

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 where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

Extremal Polyomino Chains on k-matchings and k-independent Sets

Denote by the set of polyomino chains with n squares. For any , let m _k (T _n ) and i _k (T _n ) be the number of k-matchings and k-independent sets of T _n , respectively. In this paper, we show that for any polyomino chain and any , and , with the left equalities holding for all k only if T _n =L _n , and the right equalities holding for all k only if T _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. This work is supported by NNSFC (10371102).     

On minimal energies of trees of a prescribed diameter

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let denote the set of trees on n vertices and diameter d, . Yan and Ye [Appl. Math. Lett. 18 (2005) 1046–1052] have recently determined the unique tree in with minimal energy. In this article, the trees in with second-minimal energy are characterizedAMS Subject Classification: 05C50, 05C35     

Unicycle graphs with extremal Merrifield–Simmons Index

The Merrifield–Simmons index f(G) of a (molecular) graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By we denote the set of unicycle graphs in which the length of its unique cycle is k. In this paper, we investigate the Merrifield–Simmons index f(G) for an unicycle graph G in . Unicycle graphs with the largest or smallest Merrifield–Simmons index are uniquely determined.     

On the spectral radius of graphs with connectivity at most <Emphasis Type="Italic">k</Emphasis>

In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k. We show that among those graphs, the maximal spectral radius is obtained uniquely at , which is the graph obtained by joining k edges from k vertices of K _n-1 to an isolated vertex. We also show that the spectral radius of will be very close to n − 2 for a fixed k and a sufficiently large n.     

On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)

The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, , is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, . In particular, we arrive at formulas which generalize to the case of (0 ≤ m ≤ n) the well-known Christoffel’s representation of the Legendre function of the second kind, Q _n (z). The derivatives and , all with m > n, are also evaluated.     

Extremal (<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">n</Emphasis> + 1)-graphs with respected to zeroth- order general Randić index

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. If d _v denotes the degree of the vertex v, then the zeroth-order general Randić index of the graph G is defined as , where α is a real number. We characterize, for any α, the (n,n + 1)-graphs with the smallest and greatest zeroth-order general Randić index.     

On the largest eigenvalues of trees with perfect matchings

Let λ₁ (G) and Δ (G), respectively, denote the largest eigenvalue and the maximum degree of a graph G. Let be the set of trees with perfect matchings on 2m vertices, and . Among the trees in , we characterize the tree which alone minimizes the largest eigenvalue, as well as the tree which alone maximizes the largest eigenvalue when . Furthermore, it is proved that, for two trees T ₁ and T ₂ in (m≥ 4), if and Δ (T ₁) > Δ (T ₂), then λ₁ (T ₁) > λ₁ (T ₂).     

Binding of nontarget microorganisms from food washes to anti-<Emphasis Type="Italic">Salmonella</Emphasis> and anti-<Emphasis Type="Italic">E. coli</Emphasis> O157 immunomagnetic beads: minimizing the errors of random sampling in extreme dilute systems

For most applications, 3–5 observations, or samplings (n), are utilized to estimate total aerobic plate count in an average population (μ) that is greater than about 50 cells, or colony forming units (CFU), per sampled volume. We have chosen to utilize a 6 × 6 drop plate method for bacterial colony selection because it offers the means to rapidly perform all requisite dilutions in a 96-well format and plate these dilutions on solid media using minimal materials. Besides traditional quantitative purposes, we also need to select colonies which are well-separated from each other for the purpose of bacterial identification. To achieve this goal using the drop plate format requires the utilization of very dilute solutions (μ < 10 CFUs per sampled drop). At such low CFU densities the sampling error becomes problematic. To address this issue we produced both observed and computer-generated colony count data and divided a large sample of individual counts randomly into N subsamples each with n = 2–24 observations (N × n = 360). From these data we calculated the average total mean-normalized (, n = 360) deviation of the total standard deviation (s _tot) from each jth subsample’s estimate (s _j ), which we call Δ. When either observed or computer-generated Δ values were analyzed as a function of , a set of relationships () were generated which appeared to converge at an n of about 18 observations. This finding was verified analytically at even lower CFU concentrations (). Additional experiments using the drop plate format and n = 18 samplings were performed on food samples along with most probable number (MPN) analyses and it was found that the two enumeration methods did not differ significantly. Any reference to a particular brand or company name does not constitute an endorsement of it by the U.S. Department of Agriculture over other similar brands or companies that are not mentioned.     

New upper bounds on Zagreb indices

The first Zagreb index M ₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M ₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper we obtain an upper bound on the first Zagreb index M ₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ₁), second maximum vertex degree (Δ₂) and minimum vertex degree (δ). Using this result we find an upper bound on M ₂(G). Moreover, we present upper bounds on and in terms of n,  m, Δ₁, Δ₂, δ, where denotes the complement of G.     

The first and second largest Merrifield–Simmons indices of trees with prescribed pendent vertices

The Merrifield–Simmons index of a graph G is defined as the number of subsets of the vertex set, in which any two vertices are non-adjacent, i.e., the number of independent-vertex sets of G. By T(n,k) we denote the set of trees with n vertices and with k pendent vertices. In this paper, we investigate the Merrifield–Simmons index for a tree T in T(n,k). For all trees in T(n,k), we determined unique trees with the first and second largest Merrifield–Simmons index, respectively.     

Interaction of a finite quantum system with an infinite quantum system that contains a single one-parameter eigenvalue band

Interaction of a finite quantum system that contains ρ eigenvalues and eigenstates with an infinite quantum system that contains a single one-parameter eigenvalue band is considered. A new approach for the treatment of the combined system is developed. This system contains embedded eigenstates with continuous eigenvalues , and, in addition, it may contain isolated eigenstates with discrete eigenvalues . Two ρ × ρ eigenvalue equations, a generic eigenvalue equation and a fractional shift eigenvalue equation are derived. It is shown that all properties of the system that interacts with the system can be expressed in terms of the solutions to those two equations. The suggested method produces correct results, however strong the interaction between quantum systems and . In the case of the weak interaction this method reproduces results that are usually obtained within the formalism of the perturbation expansion approach. However, if the interaction is strong one may encounter new phenomena with much more complex behavior. This is also the region where standard perturbation expansion fails. The method is illustrated with an example of a two-dimensional system that interacts with the infinite system that contains a single one-parameter eigenvalue band. It is shown that all relevant completeness relations are satisfied, however strong the interaction between those two systems. This provides a strong verification of the suggested method.     

On the Randić Index of Unicyclic Conjugated Molecules

The Randić index R(G) of a graph G is the sum of the weights of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.     

Sharp upper bounds for total π-electron energy of alternant hydrocarbons

The energy E(G) of a graph G is defined as the sum of the absolute values of all the eigenvalues of the adjacency matrix of the graph G. This quantity is used in chemistry to approximate the total π-electron energy of molecules and in particular, in case G is bipartite, alternant hydrocarbons. In this paper, we show that if G = (V ₁, V ₂; E) is a bipartite graph with edges and , then

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ω _k and normal modes of the isotopomer B are expressed in terms of the vibrational frequencies ν _i and normal modes of the parent molecule A. In those relations complete specification of the normal modes is not required. Only amplitudes at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies with respect to the DFT frequencies is on average . This error is due to the uncertainty of the input data (± 0.5 cm⁻¹) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment, one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better () than scaled DFT frequencies () which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation.