Redox‐Active‐Ligand‐Mediated Formation of an Acyclic Trinuclear Ruthenium Complex with Bridging Nitrido Ligands

Coordination of a redox‐active pyridine aminophenol ligand to Ru^II followed by aerobic oxidation generates two diamagnetic Ru^III species [ 1 a (cis) and 1 b (trans)] with ligand‐centered radicals. The reaction of 1 a / 1 b with excess NaN₃ under inert atmosphere resulted in the formation of a rare bis(nitrido)‐bridged trinuclear ruthenium complex with two nonlinear asymmetrical Ru‐N‐Ru fragments. The spontaneous reduction of the ligand centered radical in the parent 1 a / 1 b supports the oxidation of a nitride (N^3?) to half an equivalent of N₂. The trinuclear omplex is reactive toward TEMPO‐H, tin hydrides, thiols, and dihydrogen.     

Unique factorization of compositive hereditary graph properties

A graph property is any class of graphs that is closed under isomorphisms. A graph property P is hereditary if it is closed under taking subgraphs; it is compositive if for any graphs G1, G2 ∈ P there exists a graph G ∈ P containing both G1 and G2 as subgraphs. Let H be any given graph on vertices v1, . . . , vn, n ≥ 2. A graph property P is H-factorizable over the class of graph properties P if there exist P 1 , . . . , P n ∈ P such that P consists of all graphs whose vertex sets can be partitioned into n parts, possibly empty, satisfying: 1. for each i, the graph induced by the i-th non-empty partition part is in P i , and 2. for each i and j with i = j, there is no edge between the i-th and j-th parts if vi and vj are non-adjacent vertices in H. If a graph property P is H-factorizable over P and we know the graph properties P 1 , . . . , P n , then we write P = H [ P 1 , . . . , P n ]. In such a case, the presentation H[ P 1 , . . . , P n ] is called a factorization of P over P. This concept generalizes graph homomorphisms and (P 1 , . . . , P n )-colorings. In this paper, we investigate all H-factorizations of a graph property P over the class of all hered- itary compositive graph properties for finite graphs H. It is shown that in many cases there is exactly one such factorization.     

Dinuclear Palladium Complexes with Two Ligand‐Centered Radicals and a Single Bridging Ligand: Subtle Tuning of Magnetic Properties

The facile and tunable preparation of unique dinuclear [(L^?)Pd?X?Pd(L^?)] complexes (X=Cl or N₃), bearing a ligand radical on each Pd, is disclosed, as well as their magnetochemistry in solution and solid state is reported. Chloride abstraction from [PdCl( NNO^ISQ )] ( NNO^ISQ =iminosemiquinonato) with TlPF₆ results in an unusual monochlorido‐bridged dinuclear open‐shell diradical species, [{Pd( NNO ^ISQ)}₂(μ‐Cl)]⁺, with an unusually small Pd‐Cl‐Pd angle (ca. 93°, determined by X‐ray). This suggests an intramolecular d⁸–d⁸ interaction, which is supported by DFT calculations. SQUID measurements indicate moderate antiferromagnetic spin exchange between the two ligand radicals and an overall singlet ground state in the solid state. VT EPR spectroscopy shows a transient signal corresponding to a triplet state between 20 and 60 K. Complex 2 reacts with PPh₃ to generate [Pd(NNO^ISQ)(PPh₃)]⁺ and one equivalent of [PdCl( NNO^ISQ )]. Reacting an 1:1 mixture of [PdCl( NNO^ISQ )] and [Pd(N₃)( NNO^{I SQ})] furnishes the 1,1‐azido‐bridged dinuclear diradical [{Pd( NNO ^ISQ)}₂(κ¹‐N;μ‐N₃]⁺, with a Pd‐N‐Pd angle close to 127° (X‐ray). Magnetic and EPR measurements indicate two independent S=1/2 spin carriers and no magnetic interaction in the solid state. The two diradical species both show no spin exchange in solution, likely because of unhindered rotation around the Pd?X?Pd core. This work demonstrates that a single bridging atom can induce subtle and tunable changes in structural and magnetic properties of novel dinuclear Pd complexes featuring two ligand‐based radicals.     

Stochastic scheduling of parallel queues with set-up costs

We consider the problem of allocating a single server to a system of queues with Poisson arrivals. Each queue represents a class of jobs and possesses a holding cost rate, general service distribution, and a set-up cost. The objective is to minimize the expected cost due to the waiting of jobs and the switching of the server. A set-up cost is required to effect an instantaneous switch from one queue to another. We partially characterize an optimal policy and provide a simple heuristic scheduling policy. The heuristic's performance is evaluated in the cases of two and three queues by comparison with a numerically obtained optimal policy. Simulation results are provided to demonstrate the effectiveness of our heuristic over a wide range of problem instances with four queues.     

Localized Mixed‐Valence and Redox Activity within a Triazole‐Bridged Dinucleating Ligand upon Coordination to Palladium

The new dinucleating redox‐active ligand ( L^H4 ), bearing two redox‐active NNO‐binding pockets linked by a 1,2,3‐triazole unit, is synthetically readily accessible. Coordination to two equivalents of Pd^II resulted in the formation of paramagnetic (S= ) dinuclear Pd complexes with a κ²‐N,N′‐bridging triazole and a single bridging chlorido or azido ligand. A combined spectroscopic, spectroelectrochemical, and computational study confirmed Robin–Day Class II mixed‐valence within the redox‐active ligand, with little influence of the secondary bridging anionic ligand. Intervalence charge transfer was observed between the two ligand binding pockets. Selective one‐electron oxidation allowed for isolation of the corresponding cationic ligand‐based diradical species. SQUID (super‐conducting quantum interference device) measurements of these compounds revealed weak anti‐ferromagnetic spin coupling between the two ligand‐centered radicals and an overall singlet ground state in the solid state, which is supported by DFT calculations. The rigid and conjugated dinucleating redox‐active ligand framework thus allows for efficient electronic communication between the two binding pockets.     

Molecular conformation and electronic structure. II: Crystal and molecular structure ofN-(p-bromobenzylidene)-p-bromoaniline

The structure of the title compound C₁₃H₉Br₂N has been determined with MoKα₁ diffractometer data, and refined by full-matrix least squares toR = 0.061 for 888 observed reflections. The crystals are monoclinic: space groupP2₁/a,a = 24.912(13),b = 5.877(1),c = 4.046(1) Å,β = 92.42(3) °Z = 2. The uv absorption spectrum indicates a nonplanar conformation in solution. In the solid, the molecule is disordered about a crystallographic inversion center and is planar to within 0.045 Å.     

Remarks on the critical graph conjecture

The vertex-critical graph conjecture (critical graph conjecture respectively) states that every vertex-critical (critical) graph has an odd number of vertices. In this note we prove that if G is a critical graph of even order, then G has at least three vertices of less-than-maximum valency. In addition, if G is a 3-critical multigraph of smallest even order, then G is simple and has no triangles.