1,512 research outputs found

    Merging the A- and Q-spectral theories

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    Let GG be a graph with adjacency matrix A(G)A\left( G\right) , and let D(G)D\left( G\right) be the diagonal matrix of the degrees of G.G. The signless Laplacian Q(G)Q\left( G\right) of GG is defined as Q(G):=A(G)+D(G)Q\left( G\right) :=A\left( G\right) +D\left( G\right) . Cvetkovi\'{c} called the study of the adjacency matrix the AA% \textit{-spectral theory}, and the study of the signless Laplacian--the QQ\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of A(G)A\left( G\right) into Q(G)Q\left( G\right) in this paper it is suggested to study the convex linear combinations Aα(G)A_{\alpha }\left( G\right) of A(G)A\left( G\right) and D(G)D\left( G\right) defined by Aα(G):=αD(G)+(1−α)A(G),   0≤α≤1. A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1. This study sheds new light on A(G)A\left( G\right) and Q(G)Q\left( G\right) , and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.Comment: 26 page

    Spectral properties of the hierarchical product of graphs

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    The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the Susceptible-Infected-Susceptible model on a hierarchical product of two graphs

    Trace formulae for Schrodinger operators on metric graphs with applications to recovering matching conditions

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    The paper is a continuation of the study started in \cite{Yorzh1}. Schrodinger operators on finite compact metric graphs are considered under the assumption that the matching conditions at the graph vertices are of δ\delta type. Either an infinite series of trace formulae (provided that edge potentials are infinitely smooth) or a finite number of such formulae (in the cases of L1L_1 and CMC^M edge potentials) are obtained which link together two different quantum graphs under the assumption that their spectra coincide. Applications are given to the problem of recovering matching conditions for a quantum graph based on its spectrum.Comment: arXiv admin note: substantial text overlap with arXiv:1403.761

    Photoelectron spectra in an autoionization system interacting with a neighboring atom

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    Photoelectron ionization spectra of an autoionization system with one discrete level interacting with a neighbor two-level atom are discussed. The formula for long-time ionization spectra is derived. According to this formula, the spectra can be composed of up to eight peaks. Moreover, the Fano-like zeros for weak optical pumping have been identified in these spectra. The conditional ionization spectra depending on the state of the neighbor atom exhibit oscillations at the Rabi frequency. Dynamical spectral zeros occurring once per the Rabi period have been revealed in these spectra.Comment: 10 pages, 13 figure

    Essential spectra of difference operators on \sZ^n-periodic graphs

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    Let (\cX, \rho) be a discrete metric space. We suppose that the group \sZ^n acts freely on XX and that the number of orbits of XX with respect to this action is finite. Then we call XX a \sZ^n-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on lp(X)l^p(X) where XX is a \sZ^n-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on \sZ^n and their limit operators. In case XX is the set of vertices of a combinatorial graph, the graph structure defines a Schr\"{o}dinger operator on lp(X)l^p(X) in a natural way. We illustrate our approach by determining the essential spectra of Schr\"{o}dinger operators with slowly oscillating potential both on zig-zag and on hexagonal graphs, the latter being related to nano-structures
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