Pseudo-Strong Regularity Around a Set

A generalization of strong regularity around a vertex subset C of a graph Γ, which makes sense even if Γis non-regular, is studied. Such a structure appears, together with a kind of distance-regularity around C , when an spectral bound concerning the so-called predistance polynomial of C is attained. As a main consequence of these results, it is shown that a regular (connected) graph Γwith d + 1 distinct eigenvalues is distance-regular, and its distance- d graph Γ d is strongly regular with parameters a = c , if and only if the number of vertices at distance d from each vertex satisfies an expression which depends only on the order of Γand the different eigenvalues of Γ.     

Survey of ligand field parameters of strong field d complexes obtained from the g matrix

Using a newly proposed approach involving an internally consistent set of equations, the ligand field parameters Δ/ξ, V/ξ and k are obtained from literature values of the g matrix for strong field d⁵ systems of various conformations in which |Δ/ξ|≤10. Qualitative analysis of the observed results is done using the Angular Overlap Model, AOM.     

Weak Graph Map Homotopy and Its Applications∗

The authors introduce a notion of a weak graph map homotopy (they call it M-homotopy), discuss its properties and applications. They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001. The difference between them is that the weak graph map homotopy transformation is defined in terms of maps, while the graph homotopy transformation is defined by means of combinatorial operations. They discuss its advantages over the graph homotopy transformation. As its applications, they investigate the mapping class group of a graph and the 1-order MP-homotopy group of a pointed simple graph. Moreover, they show that the 1-order MP-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.     

Polariton emission in GaN microcavities

We present a detailed experimental study aimed at demonstrating the polariton nature of the photoluminescence emission in a bulk GaN microcavity grown on (111) silicon. The comparison of the photoluminescence with coincident reflectivity measurements at different temperatures shows an anticrossing behaviour with a Rabi splitting of the order of 35 meV up to room temperature. Relevant confirmations of the mixing between excitons and photons are found in the analysis of the spectral linewidth and of the time resolved kinetics.     

SC-Hamiltonian graphs and digraphs: New necessary conditions and their impacts

A graph G on n vertices is said to be separable cost constant Hamiltonian (SC-Hamiltonian) if and only if G is Hamiltonian and for any cost matrix C=(c_(i,j)) associated with G where all tours have the same cost, there exist vectors a=(a₁,…,a_n) and b=(b₁,…,b_n) such that . In this paper we show that for symmetric digraphs strong Hamiltonicity is a necessary condition for SC-Hamiltonicity. As a surprising consequence, we prove that the symmetric digraph obtained from an undirected SC-Hamiltonian graph by edge duplication need not be SC-Hamiltonian. This settles a conjecture of Kabadi and Punnen. We then show that an undirected graph on an even number of nodes having an edge that appears in every Hamiltonian cycle cannot be SC-Hamiltonian. Using this we establish that multiple subdivision of an edge need not preserve SC-Hamiltonicity, disproving a previous claim. Further, we identify other necessary conditions for SC-Hamiltonicity and obtain new classes of SC-Hamiltonian graphs.     

Analysis of longitudinal multispin orders in strongly coupled spin systems

Longitudinal multispin orders provide an effective way for measurement of scalar couplings and also to probe molecular interactions and dynamics. Analysis of longitudinal orders has been made in strongly coupled AB and ABX spin systems to determine the dependence of strong coupling parameter on these orders. Experimental and simulated spectra of various longitudinal orders are illustrated for these spin systems. This general procedure can be extended to broad range of spin systems to understand the influence of strong coupling on longitudinal orders.     

Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function f^H(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.