The local spectra of regular line graphs

The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is “seen” from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ of G is the sum, extended to all vertices, of its local multiplicities. In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the regular graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results about the number of circuits of LG

Regular quantum graphs

We introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries. The method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way incoming and outgoing channels at vertex scattering processes are connected. Symmetry properties of the quantum graph as well as its spectral statistics depend on the particular choice of permutation matrices, also called connectivity matrices, and can now be easily controlled. The method may find applications in the study of quantum random walks networks and may also prove to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure

Spectra of complex networks

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the case of networks with correlations between neighboring vertices. The tail of the density of eigenvalues $\rho(\lambda)$ at large $|\lambda|$ is related to the behavior of the vertex degree distribution $P(k)$ at large $k$. In particular, as $P(k) \sim k^{-\gamma}$, $\rho(\lambda) \sim |\lambda|^{1-2\gamma}$. We propose a simple approximation, which enables us to calculate spectra of various graphs analytically. We analyse spectra of various complex networks and discuss the role of vertices of low degree. We show that spectra of locally tree-like random graphs may serve as a starting point in the analysis of spectral properties of real-world networks, e.g., of the Internet.Comment: 10 pages, 4 figure

From quantum graphs to quantum random walks

We give a short overview over recent developments on quantum graphs and outline the connection between general quantum graphs and so-called quantum random walks.Comment: 14 pages, 6 figure

Spectra of Sparse Random Matrices

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica, managing to circumvent difficulties that have been encountered in earlier approaches along the lines first suggested in a seminal paper by Rodgers and Bray. Due attention is payed to the issue of localization. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows to unfold the total density of states into contributions coming from vertices of different local coordination.Comment: 22 papges, 8 figures (one on graph-Laplacians added), one reference added, some typos eliminate

Spectra of Modular and Small-World Matrices

We compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter- and intra-module degree distributions, subject only to the constraint of finite mean connectivities. We also evaluate spectra of a certain class of small-world matrices generated from random graphs by introducing short-cuts via additional random connectivity components. Both adjacency matrices and the associated graph Laplacians are investigated. For the Laplacians, we find Lifshitz type singular behaviour of the spectral density in a localised region of small $|\lambda|$ values. In the case of modular networks, we can identify contributions local densities of state from individual modules. For small-world networks, we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states, exhibiting only localised states in the band-gaps. Results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances, and with direct diagonalisation results.Comment: 18 pages, 5 figure

Spectral Theory of Sparse Non-Hermitian Random Matrices

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.Comment: 60 pages, 10 figure