On the digraph of a unitary matrix

Given a matrix M of size n, a digraph D on n vertices is said to be the digraph of M, when M_{ij} is different from 0 if and only if (v_{i},v_{j}) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, LD, is the digraph of a unitary matrix if and only if D is Eulerian. It follows that, if D is strongly connected and LD is the digraph of a unitary matrix then LD is Hamiltonian. We conclude with some elementary observations. Among the motivations of this paper are coined quantum random walks, and, more generally, discrete quantum evolution on digraphs.Comment: 6 page

Recognizing clique graphs of directed and rooted path graphs

AbstractWe describe characterizations for the classes of clique graphs of directed and rooted path graphs. The characterizations relate these classes to those of clique-Helly and strongly chordal graphs, respectively, which properly contain them. The characterizations lead to polynomial time algorithms for recognizing graphs of these classes

Clique graphs and Helly graphs

AbstractAmong the graphs for which the system of cliques has the Helly property those are characterized which are clique-convergent to the one-vertex graph. These graphs, also known as the so-called absolute retracts of reflexive graphs, are the line graphs of conformal Helly hypergraphs possessing a certain elimination scheme. From particular classes of such hypergraphs one can readily construct various classes G of graphs such that each member of G has its clique graph in G and is itself the clique graph of some other member of G. Examples include the classes of strongly chordal graphs and Ptolemaic graphs, respectively

The approach to criticality in sandpiles

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.Comment: 30 pages, 8 figures, long version of arXiv:0912.320

New pulsed EPR methods and their application to characterize mitochondrial complex I

Electron Paramagnetic Resonance (EPR) spectroscopy is the method of choice to study paramagnetic cofactors that often play an important role as active centers in electron transfer processes in biological systems. However, in many cases more than one paramagnetic species is contributing to the observed EPR spectrum, making the analysis of individual contributions difficult and in some cases impossible. With time-domain techniques it is possible to exploit differences in the relaxation behavior of different paramagnetic species to distinguish between them and separate their individual spectral contribution. Here we give an overview of the use of pulsed EPR spectroscopy to study the iron–sulfur clusters of NADH:ubiquinone oxidoreductase (complex I). While FeS cluster N1 can be studied individually at a temperature of 30 K, this is not possible for FeS cluster N2 due to its severe spectral overlap with cluster N1. In this case Relaxation Filtered Hyperfine (REFINE) spectroscopy can be used to separate the overlapping spectra based on differences in their relaxation behavior.Collaborative Research Centre 472 (Project P2)Collaborative Research Centre 472 (Project P15)Goethe University in Frankfurt/Main. Center for Biomolecular Magnetic Resonanc

Fast approximation of centrality and distances in hyperbolic graphs

We show that the eccentricities (and thus the centrality indices) of all vertices of a $\delta$-hyperbolic graph $G=(V,E)$ can be computed in linear time with an additive one-sided error of at most $c\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O(1)$ time an estimate $\hat{e}(v)$ of its eccentricity $ecc_G(v)$ such that $ecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree, constructible in linear time, such that for every vertex $v$ of $G$, $ecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of $G$, the smaller its eccentricity is. We also show that the distance matrix of $G$ with an additive one-sided error of at most $c'\delta$ can be computed in $O(|V|^2\log^2|V|)$ time, where $c'< c$ is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

Molecular simulations for dynamic nuclear polarization in liquids: a case study of TEMPOL in acetone and DMSO

A computational strategy for calibrating, validating and analyzing molecular dynamics (MD) simulations to predict dynamic nuclear polarization (DNP) coupling factors and relaxivities of proton spins is presented. Simulations of the polarizing agent TEMPOL in liquid acetone and DMSO are conducted at low (infinite dilution) and high (1 M) concentrations of the free radical. Because DNP coupling factors and relaxivities are sensitive to the time scales of the molecular motions, the MD simulations are calibrated to reproduce the bulk translational diffusion coefficients of the pure solvents. The simulations are then validated by comparing with experimental dielectric relaxation spectra, which report on the rotational dynamics of the molecular electric dipole moments. The analysis consists of calculating spectral density functions (SDFs) of the magnetic dipole–dipole interaction between the electron spin of TEMPOL and nuclear spins of the solvent protons. Here, MD simulations are used in combination with an analytically tractable model of molecular motion. While the former provide detailed information at relatively short spin–spin distances, the latter includes contributions at large separations, all the way to infinity. The relaxivities calculated from the SDFs of acetone and DMSO are in excellent agreement with experiments at 9.2 T. For DMSO we calculate a coupling factor in agreement with experiment while for acetone we predict a value that is larger by almost 50%, suggesting a possibility for experimental improvement