On the maximum number of diagonals of a circuit in a graph

For a graph G, let ?(G) denote the maximum number k such that G contains a circuit with k diagonals.Theorem. For any graph G with minimum valency$\text{n? 3, ?(G) ?}\text{1}\text{2}\text{(n+1)(n-2)}$.If the equality holds and G is connected, then either G is isomorphic to K_n+1 or G is separable and each of its terminal blocks is isomorphic to K_n+1, or K_n+1 with one edge subdivided.     

A model for alkali chemisorption

The linear response function in the density functional formalism, developed by Ying, Smith and Kohn, is applied to alkali chemisorption on metallic surfaces. The adion is represented by a pseudopotential perturbing the surface. Binding energies, induced dipole moments, and vibrational frequencies at the equilibrium position are first calculated for the alkalis at finite coverage, as well as, in the limit of zero coverage, on a jellium substrate. The theoretical results are in good agreement with available experimental results. The effect of electric fields on chemisorbed alkali ions is discussed with reference to interpretation of experimental results. The crystalline structure is then introduced in the form of a periodic pseudocharge density representing the ionic lattice. In this model, corrections to the quantities calculated in the jellium approximation are obtained for both low index and high index planes. Preferred adsorption sites are determined and found to be consistent with experimental findings.     

Subdirect decompositions of finite commutative semigroups

The Ponizovskîi decomposition of a finite commutative semigroup yields all shortest elementary subdirect decompositions of that semigroup.     

Photolytic rearrangement of germanium azides. Evidence for transient germa-imines

Photolysis of germanium azides gives rise to transient germa-imines which have been quantitatively trapped by pinacol. Evidence for the formation of diphenylgermylene in the photolysis of hexaphenylcyclodigermazane is also reported.