The atom assignment problem in automated de novo drug design. 2. A method for molecular graph and fragment perception

Summary If atom assignment onto 3D molecular graphs is to be optimized, an efficient scheme for placement must be developed. The strategy adopted in this paper is to analyze the molecular graphs in terms of cyclical and non-cyclical nodes; the latter are further divided into terminal and non-terminal nodes. Molecular fragments, from a fragments database, are described in a similar way. A canonical numbering scheme for the fragments and the local subgraph of the molecular graph enables fragments to be placed efficiently onto the molecular graph. Further optimization is achieved by placing similar fragments into bins using a hashing scheme based on the canonical numbering. The graph perception algorithm is illustrated in detail.     

Synthesis of new bicyclic lactam peptidomimetics by ring-closing metathesis reactions

An efficient and versatile synthetic method for the preparation of new fused bicyclic lactams 3a and 3b is described. The spirane cyclopentane nucleus was easily installed by diallylation of the pyroglutamate derivative 18 followed by ring-closing metathesis (RCM). A more practical and stereoselective method for the allylation of the α-methoxy carbamate 21, involving the use of InCl₃ as a Lewis acid, was developed. In the crucial coupling reaction of the diastereomeric mixture of cis- and trans-pirrolidine derivatives 5a and 5b with N-Cbz vinyl phenylalanine only the cis isomer was found to react. An RCM reaction on the dipeptides 25a and 25b followed by catalytic hydrogenation, gave the final epimeric bicyclic lactams 3a and 3b. The same synthetic sequence on the model compound 7, lacking the spiro cyclopentane nucleus, is also reported.     

Phosphorus–nitrogen compounds. Part VI. Aminolysis of octachlorocyclotetraphosphazatetraene and the crystal structure of 2-trans-6-bis(n-propylamino)-2,4,4,6,8,8-hexakis-tert-butylaminocyclo-2λ, 4λ, 6λ, 8λ-tetraphosphazatetraene

The reactions of 2-trans-6-N₄P₄(NHPrⁿ)₂Cl₆ (2), which was obtained from N₄P₄Cl₈ (1) and n-propylamine, with pyrrolidine and t-butylamine in different solvents have been studied. Compound (2) gave two different products, namely monocyclic (3 and 5) and bicyclic (4 and 6) phosphazenes. Compounds (2–6) have been characterized by elemental analysis, IR, ¹H-, ¹³C-, ³¹P NMR, HETCOR and MS and the structure of compound (5) has been examined crystallographically. The bicyclic phosphazene (6) is the first exciting example of bicyclic phosphazenes containing chlorine atoms, in the literature. The formation mechanisms of bicyclic phosphazenes are re-considered by taking into account the synthesis of compound (6), which contains three stereogenic phosphorus atoms. Compound (5) crystallizes in the monocyclic space group P2₁/n with a=13.974(2), b=17.836(5), and c=18.683(4) Å, β=98.50(1)°, V=4605.4(2) Å³, Z=4 and D_x=1.051 g cm^−3. It consists of a non-centrosymmetric, non-planar phosphazene ring in a saddle conformation, with two n-propylamino (in 2-trans-6 positions) and six bulky t-butylamino side groups. The bulky substituents are instrumental in determining the molecular geometry.     

Further analysis of the number of spanning trees in circulant graphs

Let 1?s₁<s₂<?<s_k?⌊n/2⌋ be given integers. An undirected even-valent circulant graph, has n vertices 0,1,2,…, n-1, and for each and j(0?j?n-1) there is an edge between j and . Let stand for the number of spanning trees of . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X. Yong, M. Golin, [The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]], is that where a_n satisfies a linear recurrence relation of order 2^s_k-1. And, most recently, for odd-valent circulant graphs, a nice investigation on the number a_n is [X. Chen, Q. Lin, F. Zhang, The number of spanning trees in odd-valent circulant graphs, Discrete Math. 282 (2004) 69-79].In this paper, we explore further properties of the numbers a_n from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for a_n, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we ‘answer’ the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.     

A Degree Constraint for Uniquely Hamiltonian Graphs

A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log₂n+2 where c=(log₂3−1)⁻¹≈1.71 thereby improving a bound given by Bondy and Jackson [3].     

Minimal Connected τ-Critical Hypergraphs

A hypergraph is τ-critical if τ(−{E})<τ() for every edge E ∈ , where τ() denotes the transversal number of . We show that if is a connected τ-critical hypergraph, then −{E} can be partitioned into τ()−1 stars of size at least two, for every edge E ∈ . An immediate corollary is that a connected τ-critical hypergraph has at least 2τ()−1 edges. This extends, in a very natural way, a classical theorem of Gallai on colour-critical graphs, and is equivalent to a theorem of Füredi on t-stable hypergraphs. We deduce a lower bound on the size of τ-critical hypergraphs of minimum degree at least two.     

Connectivity of Cartesian product graphs

Use v_i,κ_i,λ_i,δ_i to denote order, connectivity, edge-connectivity and minimum degree of a graph G_i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are κ(G₁×G₂)?κ₁+κ₂ and λ(G₁×G₂)?λ₁+λ₂. This paper improves these results by proving that κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁} and λ(G₁×G₂)=min{δ₁+δ₂,λ₁v₂,λ₂v₁} if G₁ and G₂ are connected undirected graphs; κ(G₁×G₂)?min{κ₁+δ₂,κ₂+δ₁,2κ₁+κ₂,2κ₂+κ₁} if G₁ and G₂ are strongly connected digraphs. These results are also generalized to the Cartesian products of connected graphs and n strongly connected digraphs, respectively.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.