On the Total Domination Number of Cartesian Products of Graphs

The most famous open problem involving domination in graphs is Vizings conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for total domination. In particular, we prove that the product of the total domination numbers of any nontrivial tree and any graph without isolated vertices is at most twice the total domination number of their Cartesian product, and we characterize the extremal graphs.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Operator algebras and a theorem of Dieudonne

LetA be aC*-algebra with second dualA″. Let (φ _n)(n=1,...) be a sequence in the dual ofA such that limφ _n(a) exists for eacha εA. In general, this does not imply that limφ _n(x) exists for eachx εA″. But if limφ _n(p) exists whenever p is the range projection of a positive self-adjoint element of the unit ball ofA, then it is shown that limφ _n(x) does exist for eachx inA″. This is a non-commutative generalisation of a celebrated theorem of Dieudonné. A new proof of Dieudonné’s theorem, for positive measures, is given here. The proof of the main result makes use of Dieudonné’s original theorem.     

Substrate mimetics-specific peptide ligases: studies on the synthetic utility of a zymogen and zymogen-like enzymes

Although proteases are capable of synthesizing peptide bonds, they are not proficient at peptide fragment ligation. Further manipulations are needed to shift the native enzyme activity from the cleavage to the synthesis of peptides. This account reports on the synthetic potential of nonactivatable trypsinogen and zymogen-like enzymes designed to minimize proteolytic side reactions during peptide synthesis.     

Metal-induced tautomerization of p- to o-quinone compounds: experimental evidence from CuI and ReI complexes of azophenine and DFT studies

Azophenine (7,8-diphenyl-2,5-bis(phenylamino)-p-quinonediimine, L(p)) reacts with [Cu(PPh3)4](BF4) or [Re(CO)(5)Cl] to yield the (Ph3P)(2)Cu(+) or [(OC)(3)ClRe] complex of the tautomeric form 7,8-diphenyl-4,5-bis(phenylamino)-o-quinonediimine, L(o), as evident from structure determinations and from very intense metal-to-ligand charge transfer (MLCT) transitions in the visible region. Time-dependent DFT (TD-DFT) calculations on model complexes [(N intersection N)Re(CO)(3)Cl] confirm the spectroscopic results, showing considerably higher oscillator strengths of the MLCT transition for the o-quinonediimine complexes in comparison to compounds with N intersection N=1,4-dialkyl-1,4-diazabutadiene. The complexes are additionally stabilized through hydrogen bonding between two now ortho-positioned NHPh substituents and one fluoride of the BF(4) (-) anion (Cu complex) or the chloride ligand (Re complex). DFT Calculations on the model ligands p-quinonediimine or 2,5-diamino-p-quinonediimine and their ortho-quinonoid forms with and without Li(+) or Cu(+) are presented to discuss the relevance for metal-dependent quinoproteins.     

A Coordination-Induced 1,4→1,2-Quinonediimine Isomerization

Hitherto unused in coordination compounds , the para-quinonoid azophenine (p-ap) has now been applied for the synthesis of [Cu(PPh₃)₂(o-ap)](BF₄) ( 1 ), in which the ligand exists as its ortho-quinonoid tautomer. Additional stabilization of this metal–π donor/ligand–π acceptor arrangement occurs through chelate-like hydrogen bonding between the NH functionalities, which are both now ortho-positioned, and the BF₄⁻ counterion.