12,764 research outputs found

    On the Signed 22-independence Number of Graphs

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    In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area

    A Survey on Alliances and Related Parameters in Graphs

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    In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, α\alpha-domination, α\alpha-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global rr-alliances in graphs.We also propose a new framework, called (global) (D,O)(D,O)-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

    Approximate Zero Modes for the Pauli Operator on a Region

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    Let PΩ,tA\mathcal{P}_{\Omega,tA} denoted the Pauli operator on a bounded open region ΩR2\Omega\subset\mathbb{R}^2 with Dirichlet boundary conditions and magnetic potential AA scaled by some t>0t>0. Assume that the corresponding magnetic field B=curlAB=\mathrm{curl}\,A satisfies BLlogL(Ω)Cα(Ω0)B\in L\log L(\Omega)\cap C^\alpha(\Omega_0) where α>0\alpha>0 and Ω0\Omega_0 is an open subset of Ω\Omega of full measure (note that, the Orlicz space LlogL(Ω)L\log L(\Omega) contains Lp(Ω)L^p(\Omega) for any p>1p>1). Let NΩ,tA(λ)\mathsf{N}_{\Omega,tA}(\lambda) denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula NΩ,tA(λ(t))=t2πΩB(x)dx  +o(t) \mathsf{N}_{\Omega,tA}(\lambda(t))=\frac{t}{2\pi}\int_{\Omega}\lvert B(x)\rvert\,dx\;+o(t) as t+t\to+\infty, whenever λ(t)=Cectσ\lambda(t)=Ce^{-ct^\sigma} for some σ(0,1)\sigma\in(0,1) and c,C>0c,C>0. The corresponding eigenfunctions can be viewed as a localised version of the Aharonov-Casher zero modes for the Pauli operator on R2\mathbb{R}^2.Comment: 28 pages; for the sake of clarity the main results have been reformulated and some minor presentational changes have been mad
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