Topic
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.
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TL;DR: In this article, a method based on spectral graph theory is used to determine whether a portfolio is balanced or not, and combine this with spectral clustering to propose a strategy to build a balanced portfolio with hedging assets.
Abstract: A portfolio associated with a balanced signed graph that contains both positive and negative edges is more predictable and risk-averse, and is therefore likely to require less turnover. In this paper, we use a method based on spectral graph theory to determine whether a portfolio is balanced or not. Moreover, we combine this with spectral clustering to propose a strategy to build a balanced portfolio with hedging assets. This is applied to stocks listed on the Brazil Stock Exchange.
4 citations
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TL;DR: This work shows that a corresponding dual problem allows us to view eigenvectors to optimized eigenvalues as graph realizations in Euclidean space, whose structure is tightly linked to the separator structure of the g...
Abstract: Extremal eigenvalues and eigenvectors of the Laplace matrix of a graph form the core of many bounds on graph parameters and graph optimization problems. For example, the value of a uniform sparsest cut is bounded from below and above by the second smallest and the largest eigenvalue of the weighted Laplacian divided by the number of nodes of the graph. Minimizing the difference between maximum and second smallest eigenvalue over edge weighted Laplacians of a graph reduces the size of this interval. We study this problem of minimizing the spectral width for its own sake with the goal of advancing the understanding of connections between structural properties of the graph and the corresponding eigenvectors and eigenvalues. Building on previous work where these eigenvalues were investigated separately, we show that a corresponding dual problem allows us to view eigenvectors to optimized eigenvalues as graph realizations in Euclidean space, whose structure is tightly linked to the separator structure of the g...
4 citations
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TL;DR: The spectral sheaf theory as discussed by the authors is an extension of spectral graph theory to cellular sheaves, which can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graphs.
Abstract: This paper outlines a program in what one might call spectral sheaf theory --- an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, sparsification, effective resistance, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.
4 citations
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07 Jul 2010
TL;DR: In this paper, a unified approach to study convergence and stochastic stability of continuous time consensus protocols (CPs) is presented, which applies to networks with directed information flow; both cooperative and non-cooperative interactions; networks under weak stochastically forcing; and those whose topology and strength of connections may vary in time.
Abstract: A unified approach to studying convergence and stochastic stability of continuous time consensus protocols (CPs) is presented in this work. Our method applies to networks with directed information flow; both cooperative and noncooperative interactions; networks under weak stochastic forcing; and those whose topology and strength of connections may vary in time. The graph theoretic interpretation of the analytical results is emphasized. We show how the spectral properties, such as algebraic connectivity and total effective resistance, as well as the geometric properties, such the dimension and the structure of the cycle subspace of the underlying graph, shape stability of the corresponding CPs. In addition, we explore certain implications of the spectral graph theory to CP design. In particular, we point out that expanders, sparse highly connected graphs, generate CPs whose performance remains uniformly high when the size of the network grows unboundedly. Similarly, we highlight the benefits of using random versus regular network topologies for CP design. We illustrate these observations with numerical examples and refer to the relevant graph-theoretic results. Keywords: consensus protocol, dynamical network, synchronization, robustness to noise, algebraic connectivity, effective resistance, expander, random graph
4 citations
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TL;DR: In this article, the eigenvalues of a class of fullerene graphs are investigated, where the graph is a 3-connected graph with 12 pentagonal faces and n/2 -10 hexagonal faces.
Abstract: The eigenvalues of a graph is the root of its characteristic polynomial. A fullerene F is a 3- connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n is the number of vertices of F. In this paper we investigate the eigenvalues of a class of fullerene graphs.
4 citations