An iterative approach to graph irregularity strength

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s(T) for trees T in which every two vertices of degree not equal to two are at distance at least eight.     

Orders in rings without identity

G. Lallement [5] proved that every idem potent congruence class of a regular semigroup contains an idem potent. P. Edwards [4] generalized this property of congruences to eventually regular semigroups. Using the natural partial order of the semigroup (see [6]) a weakened version of this result will be proved for the more general class of E-inversive semigroups. But for particular congruences the original result of Lallement still holds for every E-inversive semigroup. Finally, conditions for a congruence on a general semigroup (with E(S) a subsemigroup, resp.) are given, which ensure that Lallement's result holds.     

Minimum degree and disjoint cycles in generalized claw-free graphs

For s≥3$s\ge 3$ a graph is _K1,s$K_{1,s}$-free if it does not contain an induced subgraph isomorphic to _K1,s$K_{1,s}$. Cycles in _K1,3$K_{1,3}$-free graphs, called claw-free graphs, have been well studied. In this paper we extend results on disjoint cycles in claw-free graphs satisfying certain minimum degree conditions to _K1,s$K_{1,s}$-free graphs, normally called generalized claw-free graphs. In particular, we prove that if G$G$ is _K1,s$K_{1,s}$-free of sufficiently large order n=3k$n=3k$ with δ(G)≥n/2+c$\delta \left(G\right)\ge n/2+c$ for some constant c=c(s)$c=c\left(s\right)$, then G$G$ contains k$k$ disjoint triangles. Analogous results with the complete graph _K3$K_3$ replaced by a complete graph _Km$K_m$ for m≥3$m\ge 3$ will be proved. Also, the existence of 2$2$-factors for _K1,s$K_{1,s}$-free graphs with minimum degree conditions will be shown.     

A definition for the covalent and ionic bond index in a molecule

Formulae for hermitian operators representing covalent, ionic, and total bond indices are derived. The eigenstates of these operators come in pairs, and can be considered as bonding, anti-bonding and lone-pair orbitals. The form of these operators is derived by generalising the rule that the bond order be defined as the net number of bonding electron pairs. The percentage of covalency and ionicity of a chemical bond may be obtained, and bond indices can also be defined between groups of atoms. The calculation of the bond indices depends only on the electron density operator, and certain projection operators used to represent each atom in the molecule. Bond indices are presented for a series of first and second row hydrides and fluorides, hydrocarbons, a metal complex, a Diels–Alder reaction and a dissociative reaction. In general the agreement between the bond indices is in accord with chemical intuition. The bond indices are shown to be stable to basis set expansion.     

Synthesis and solid-state dynamics of molecular dirotors

We report the design, synthesis, thermal properties, and dynamic NMR characterization of four new molecular dirotors, each containing two rotary 1,4-diethynyl phenylene units, a bridging 1,3-bis(diphenylmethyl) benzene stator, and two capping triphenylmethyl groups. These compounds are variations of the well-known wheel and axle structures. Ambient temperature quadrupolar echo ²H NMR spectra of phenylene-deuterated samples revealed line shapes consistent with dynamic processes having exchange rates >10⁸ s⁻¹, and more than the two minima typically observed for most phenylene rotators. Reasonable line-shape simulations were obtained with a model that involves four minima related by angular displacements of 0°, 30°, 180°, and 230°. Cooling the samples to 260 K revealed spectra consistent with samples that contain two or more components. These results indicate that novel wheel and axle structures do create a certain amount of free volume that allows for the desired mobility of the rotating units.     

Decongestion of methylene spectra in biological and non-biological systems using picosecond 2DIR spectroscopy measuring electron-vibration–vibration coupling

Methylene is found in the repeat units of many polymers including proteins. In some cases it appears to be a useful reporter of variation in local environment whilst in other contexts average behaviour seems to dominate. In this paper we apply a particular 2DIR technique to a range of systems containing methylene groups, showing that mode frequencies, linewidths and splittings can be easily extracted even when the infrared absorption bands are too congested to allow reliable analysis. 2DIR spectra of polyethylene and several liquid alkanes are compared and it is shown for the case of l-arginine that the methylene scissor modes are split and that this can be resolved by tracking the 2DIR spectrum as a function of time. Calculations from first principles reveal that for most of the methylene modes studied, electrical anharmonicity is the dominant contributor to the 2DIR cross-peak intensity, with the mechanical anharmonicity making only a small contribution.     

Restriction semigroups and $$\lambda $$-Zappa-Szép products

The aim of this paper is to study \(\lambda \)-semidirect and \(\lambda \)-Zappa-Szép products of restriction semigroups. The former concept was introduced for inverse semigroups by Billhardt, and has been extended to some classes of left restriction semigroups. The latter was introduced, again in the inverse case, by Gilbert and Wazzan. We unify these concepts by considering what we name the scaffold of a Zappa-Szép product \(S\bowtie T\) where S and T are restriction. Under certain conditions this scaffold becomes a category. If one action is trivial, or if S is a semilattice and T a monoid, the scaffold may be ordered so that it becomes an inductive category. A standard technique, developed by Lawson and based on the Ehresmann-Schein-Nambooripad result for inverse semigroups, allows us to define a product on our category. We thus obtain restriction semigroups that are \(\lambda \)-semidirect products and \(\lambda \)-Zappa-Szép products, extending the work of Billhardt and of Gilbert and Wazzan. Finally, we explicate the internal structure of \(\lambda \)-semidirect products.     

Comparison of basis set effects and the performance of ab initio and DFT methods for probing equilibrium fluctuations

The electronic absorption and emission spectra of large molecules reflect the extent and timescale of electron-vibration coupling and therefore the extent and timescale of relaxation/reorganization in response to a perturbation. In this paper, we present a comparison of the calculated absorption and emission spectra of NADH in liver alcohol dehydrogenase (LADH), using quantum mechanical/molecular mechanical methods, in which we vary the QM component. Specifically, we have looked at the influence of basis set (STO-3G, 3-21G*, 6-31G*, CC-pVDZ, and 6-311G**), as well as the influence of applying the DFT TD-B3LYP and ab initio TD-HF and CIS methods to the calculation of absorption/emission spectra and the reorganization energy (Stokes shift). The ab initio TD-HF and CIS methods reproduce the experimentally determined Stokes shift and spectral profiles to a high level of agreement, while the TD-B3LYP method significantly underestimates the Stokes shift, by 45%. We comment on the origin of this problem and suggest that ab initio methods may be naturally more suited to predicting molecular behavior away from equilibrium geometries.