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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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Journal ArticleDOI
TL;DR: The study on neighborhood may be used to represent graph in computer algorthim, neighborhood are used to determine the clustering cofficient of graph and adjacency martix is useful in computer application.
Abstract: The present investigation is concerned with zero divisor graph of direct Product of finite commutative rings and to give some new ideas about its corresponding adjacency matrix. In the first section of the paper, we study about neighborhood set of the zero divisor graph of direct product over finite commutative rings. In the second section we discussed some examples of these ring. Finally, some surprsing results (regarding to the adjacency matrix) and theorems also estabilised. The study on neighborhood may be used to represent graph in computer algorthim, neighborhood are used to determine the clustering cofficient of graph and adjacency martix is useful in computer application.

14 citations

Proceedings ArticleDOI
10 May 2021
TL;DR: In this paper, the problem of network throughput optimization of an IRS-assisted multi-hop MEC network is investigated, in which the phase shifts of the RIS and the resource allocation of the relays need to be jointly optimized.
Abstract: The performance of mobile edge computing (MEC) depends critically on the quality of the wireless channels. From this viewpoint, the recently advocated intelligent reflecting surface (IRS) technique that can proactively reconfigure wireless channels is anticipated to bring unprecedented performance gain to MEC. In this paper, the problem of network throughput optimization of an IRS-assisted multi-hop MEC network is investigated, in which the phase-shifts of the IRS and the resource allocation of the relays need to be jointly optimized. However, due to the coupling among the transmission links of different hops caused by the utilization of the IRS and the complicated multi-hop network topology, it is difficult to solve the considered problem by directly applying existing optimization techniques. Fortunately, by exploiting the underlying structure of the network topology and spectral graph theory, it is shown that the network throughput can be well approximated by the second smallest eigenvalue of the network Laplacian matrix. This key finding allows us to develop an effective iterative algorithm for solving the considered problem. Numerical simulations are performed to corroborate the effectiveness of the proposed scheme.

14 citations

Journal ArticleDOI
TL;DR: In this article, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow in complex realworld phenomena across a wide range of scales, from aviation and Internet traffic to signal propagation in electronic and gene regulatory circuits.
Abstract: Complex real-world phenomena across a wide range of scales, from aviation and Internet traffic to signal propagation in electronic and gene regulatory circuits, can be efficiently described through dynamic network models. In many such systems, the spectrum of the underlying graph Laplacian plays a key role in controlling the matter or information flow. Spectral graph theory has traditionally prioritized analyzing unweighted networks with specified adjacency properties. Here, we introduce a complementary framework, providing a mathematically rigorous weighted graph construction that exactly realizes any desired spectrum. We illustrate the broad applicability of this approach by showing how designer spectra can be used to control the dynamics of various archetypal physical systems. Specifically, we demonstrate that a strategically placed gap induces generalized chimera states in Kuramoto-type oscillator networks, tunes or suppresses pattern formation in a generic Swift-Hohenberg model, and leads to persistent localization in a discrete Gross-Pitaevskii quantum network. Our approach can be generalized to design continuous band gaps through periodic extensions of finite networks.

14 citations

Posted Content
TL;DR: In this article, a framework based on spectral graph theory was proposed to capture the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance.
Abstract: We present a framework based on spectral graph theory that captures the interplay among network topology, system inertia, and generator and load damping in determining the overall grid behavior and performance. Specifically, we show that the impact of network topology on a power system can be quantified through the network Laplacian eigenvalues, and such eigenvalues determine the grid robustness against low frequency disturbances. Moreover, we can explicitly decompose the frequency signal along scaled Laplacian eigenvectors when damping-inertia ratios are uniform across buses. The insight revealed by this framework partially explains why load-side participation in frequency regulation not only makes the system respond faster, but also helps lower the system nadir after a disturbance. Finally, by presenting a new controller specifically tailored to suppress high frequency disturbances, we demonstrate that our results can provide useful guidelines in the controller design for load-side primary frequency regulation. This improved controller is simulated on the IEEE 39-bus New England interconnection system to illustrate its robustness against high frequency oscillations compared to both the conventional droop control and a recent controller design.

14 citations

Posted Content
TL;DR: In this article, a framework for defining a diffusion operator on a directed hypergraph with stationary vertices is presented, which is general enough for the following two applications: quadratic optimization and semi-supervised learning.
Abstract: In spectral graph theory, the Cheeger's inequality gives upper and lower bounds of edge expansion in normal graphs in terms of the second eigenvalue of the graph's Laplacian operator. Recently this inequality has been extended to undirected hypergraphs and directed normal graphs via a non-linear operator associated with a diffusion process in the underlying graph. In this work, we develop a unifying framework for defining a diffusion operator on a directed hypergraph with stationary vertices, which is general enough for the following two applications. 1. Cheeger's inequality for directed hyperedge expansion. 2. Quadratic optimization with stationary vertices in the context of semi-supervised learning. Despite the crucial role of the diffusion process in spectral analysis, previous works have not formally established the existence of the corresponding diffusion processes. In this work, we give a proof framework that can indeed show that such diffusion processes are well-defined. In the first application, we use the spectral properties of the diffusion operator to achieve the Cheeger's inequality for directed hyperedge expansion. In the second application, the diffusion operator can be interpreted as giving a continuous analog to the subgradient method, which moves the feasible solution in discrete steps towards an optimal solution.

14 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981