Large families of laplacian isospectral graphs

Two graphs are isomorphic only if they are Laplacian isospectral, that is, their Laplacian matrices share the same multiset of eigenvalues. Large families of nonisomorphic Laplacian isospectral graphs are exhibited for which the common multiset of eigenvalues consists entirely of integers.     

On Vertex Connectivity and Absolute Algebraic Connectivity for Graphs

Let G be a graph on n vertices with vertex connectivity v with 1 h v h n m 2. We produce an attainable upper bound on the absolute algebraic connectivity of G in terms of n and v .     

Bond functions and the topological properties of the bonds

The effect of bond functions on the topological properties of the bonds is studied by performing Hartree-Fock calculations for O₂, Cl₂, F₂, CN^–, N₂ and LiCl molecules. The results show that bond functions increse the degree of covalency of the bonds and the electronic density at the (3, –1) bond critical point. This effect is quite noticeable for intermediate interactions molecules (F₂ and CN^–) where a is obtained instead of a positive value. The optimal position of the bond functions seems to be at or near the (3, –1) bond critical point.     

On normalized Laplacians,multiplicative degree-Kirchhoff indices,and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives

Let be the molecular graph of the linear [n] phenylene with n hexagons and n − 1 squares, and let be the graph obtained by attaching four-membered rings to the terminal hexagons of . In this article, the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 3n is determined. An explicit closed-form formula of the multiplicative degree-Kirchhoff index (respectively the number of spanning trees) of is derived. Similarly, explicit closed-form formulas of the multiplicative degree-Kirchhoff index and the number of spanning trees of are obtained. It is interesting to see that the multiplicative degree-Kirchhoff index of (respectively ) is approximately to one half of its Gutman index.     

The molecular and electronic structure features of silatranes, germatranes, and their carbon analogs

Molecular geometries of fifty-six metallatranes N(CH₂CH₂Y)₃M-X and fifty-six carbon analogs HC(CH₂CH₂Y)₃M-X (M = Si, Ge; X = H, Me, OH, F; Y = CH₂, O, NH, NMe, NSiMe₃, PH, S) were optimized by the DFT method. Correlations between changes in the bond orbital populations, electron density ρ(r), electron density laplacian ∇²ρ(r), |λ₁|/λ₃ ratio, electronic energy density E(r), bond lengths, and displacement of the central atom from the plane of three equatorial substituents and the nature of substituents X and Y were studied. As the number of electronegative substituents at the central atom increases, the M←N, M-X, and M-Y bond lengths decrease, while the M←N bond strength and the electron density at critical points of the M←N, M-X, and M-Y bonds increase. An increase in electronegativity of a substituent (X or Y) is accompanied by a decrease in the ionicities of the other bonds (M-X, M-Y, and M←N) formed by the central atom (Si, Ge). A new molecular orbital diagram for bond formation is proposed, which takes into account the interaction of all five substituents at the central atom (M = Si, Ge) in atrane molecules. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 448–460, March, 2006.     

Asymptotic behaviour of the phase in non-smooth obstacle scattering

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle , where Ω is a bounded open subset of ℝ ⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω _e =ℝ ⁿ . We can prove that if Ω satisfies a certain geometrical condition, then

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

Estimate of the second term in the spectral asymptotic of Cauchy transform

We give the best possible (respecting to order) estimate of the remainder term in spectral asymptotics of the Cauchy operator and the logarithmic potential type operator. Also, we state a conjecture on the exact value of the remainder term.     

Second order problems with functional conditions including Sturm–Liouville and multipoint conditions

In this paper we study the solvability of equations of the form ‐(d/dt)φ (t, u, u (t), u ′(t)) = f (t, u, u (t), u ′(t)) for a.e. t ∈ I = [a, b ], together with functional‐boundary conditions which cover, amongst others, Sturm–Liouville and multipoint boundary data as particular cases. Our approach uses upper and lower solutions together with growth restrictions of Nagumo's type. An example is presented to show the applicability of the obtained results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on extinction for fast diffusive p‐Laplacian with sources

In this note we illuminate that the small condition on initial data u₀ in Theorem 4.1 of Yin and Jin (Math. Meth. Appl. Sci. 2007; 30 (10):1147–1167) can be removed for the case p?1<q<1. Precise decay estimates of solution are also obtained. Copyright © 2007 John Wiley & Sons, Ltd.