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Search: All articles in the CMB digital archive with keyword cycles

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1. CMB 2013 (vol 57 pp. 413)

Samei, Karim
 On the Comaximal Graph of a Commutative Ring Let \$R\$ be a commutative ring with \$1\$. In [P. K. Sharma, S. M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176(1995) 124-127], Sharma and Bhatwadekar defined a graph on \$R\$, \$\Gamma(R)\$, with vertices as elements of \$R\$, where two distinct vertices \$a\$ and \$b\$ are adjacent if and only if \$Ra + Rb = R\$. In this paper, we consider a subgraph \$\Gamma_2(R)\$ of \$\Gamma(R)\$ which consists of non-unit elements. We investigate the behavior of \$\Gamma_2(R)\$ and \$\Gamma_2(R) \setminus \operatorname{J}(R)\$, where \$\operatorname{J}(R)\$ is the Jacobson radical of \$R\$. We associate the ring properties of \$R\$, the graph properties of \$\Gamma_2(R)\$ and the topological properties of \$\operatorname{Max}(R)\$. Diameter, girth, cycles and dominating sets are investigated and the algebraic and the topological characterizations are given for graphical properties of these graphs. Keywords:comaximal, Diameter, girth, cycles, dominating setCategory:13A99

2. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
 The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology We prove that for a generic hypersurface in \$\mathbb P^{2n+1}\$ of degree at least \$2+2/n\$, the \$n\$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing. Keywords:Noether--Lefschetz, algebraic cycles, Picard numberCategories:14C15, 14C25