Let $kK_{r+1}$ be the graph consisting of $k$ vertex-disjoint copies of the complete graph $K_{r+1}$. Moon [Canad. J. Math. 20 (1968) 95--102] and Simonovits [Theory of Graphs (Proc. colloq., Tihany, 1996)] independently showed that if $n$ is sufficiently large, then the join of a complete graph $K_{k-1}$ and an $r$-partite Turán graph $T_{n-k+1,r}$ is the unique extremal graph for $kK_{r+1}$. In this paper we consider the graph which has the maximum spectral radius among all graphs without $k$ disjoint cliques. We show that if $G$ attains the maximum spectral radius over all $n$-vertex $kK_{r+1}$-free graphs for sufficiently large $n$, then $G$ is isomorphic to the join of a complete graph $K_{k-1}$ and an $r$-partite Turán graph $T_{n-k+1,r}$.
An unprecedented cyclization reaction of donor−acceptor oxiranes with N,N′-disubstituted thioureas to construct trans-dihydropyrimidines is presented. Preliminary reaction mechanism studies demonstrated that the reaction underwent sequential cycloaddition/amine ester exchange/oxygen−sulfur exchange/desulfuration/Michael addition process. A wide range of trans-dihydropyrimidines were produced with high yields up to 94% by using this method.
The GAF domain of phytochrome is essential for photoconversion and signal transduction. In gymnosperms, it exists in all members of the phytochrome family that experience gene duplication. Maximum-likelihood models of codon substitution can provide a framework for constructing likelihood ratio tests of changes in selective pressure and make clear predictions about patterns of genetic change following gene duplication. In this study, 68 gymnosperm GAF sequences were analyzed to identify lineages and sites under positive selection. Our results indicate that (1) positive selection at a few sites (3.6%), rather than relaxation of selective constraints, has played a major role in the evolution of the gymnosperm GAF domain; (2) strong positive selective pressure tends to occur in the recent PHYP lineages of cogeneric species, but is absent in old lineages consisting of distantly related species; and (3) the selective pressure indicated by the omega ratio varies greatly among lineages and sites in the GAF domain.
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