Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Trace formulas for general Hermitian matrices: unitary scattering approach and periodic orbits on an associated graph

Abstract: Two trace formulas for the spectra of arbitrary Hermitian matrices are derived by transforming the given Hermitian matrix $H$ to a unitary analogue. In the first type the unitary matrix is $e^{i(\lambda\II - H)}$ where $\lambda$ is the spectral parameter. The new feature is that the spectral parameter appears in the final form as an argument of Eulerian polynomials -- thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin's theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

Using Treaps for Optimization of Graph Storage

Abstract: Adjacency matrix is an effective technique used to represent a graph or a Social network comprising of large number of vertices and edges. The intent is of this paper is to optimize the graph storage and mapping without using a large adjacency matrix to represent a large graph. A special data structure Treap, a combination of binary search tree and heaps has been used as a replacement to a large adjacency matrix. It has been experimentally evaluated that the proposed approach significantly improves the space occupied by adjacency matrix and helps the graph to grow dynamically without affecting the current data structure.

Graph theory in the analysis of nonlinear systems

Abstract: Graph techniques together with the nonharmonic series representation of stationary Gaussian signals can be used in the analysis of a wide class of nonlinear systems. This point is illustrated by the derivation of the moments of a nenlinear functional of a Gaussian signal. A graphical method is developed for the derivation of the time average of products of functions of a stationary Gaussian signal without resort to any probability theory considerations. Consequently, the derivation of the moments of the functional reduces to a graph enumeration procedure and the computation of a finite number of integrals.

Accelerated filtering on graphs using Lanczos method

Abstract: Signal-processing on graphs has developed into a very active field of research during the last decade. In particular, the number of applications using frames constructed from graphs, like wavelets on graphs, has substantially increased. To attain scalability for large graphs, fast graph-signal filtering techniques are needed. In this contribution, we propose an accelerated algorithm based on the Lanczos method that adapts to the Laplacian spectrum without explicitly computing it. The result is an accurate, robust, scalable and efficient algorithm. Compared to existing methods based on Chebyshev polynomials, our solution achieves higher accuracy without increasing the overall complexity significantly. Furthermore, it is particularly well suited for graphs with large spectral gaps. 1

On the class of graphs with strong mixing properties

Abstract: We study three mixing properties of a graph: large algebraic connectivity, large Cheeger constant (isoperimetric number) and large spectral gap from 1 for the second largest eigenvalue of the transition probability matrix of the random walk on the graph. We prove equivalence of this properties (in some sense). We give estimates for the probability for a random graph to satisfy these properties. In addition, we present asymptotic formulas for the numbers of Eulerian orientations and Eulerian circuits in an undirected simple graph.