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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Journal Article•
TL;DR: The comparison shows that the performances of unnormalized and normalized symmetric graph Laplacian matrices are similar, and they are better than that of normalized random walk graph LaPLacian matrix.
Abstract: Detecting protein complexes is an important way to discover the relationship between network topological structure and its functional features in protein-protein interaction (PPI) network. The spectral clustering method is a popular approach. However, how to select its optimal Laplacian matrix is still an open problem. Here, we analyzed the performances of three graph Laplacian matrices (unnormalized symmetric graph Laplacians,, normalized symmetric graph Laplacians and normalized random walk graph Laplacians, respectively) in yeast PPI network. The comparison shows that the performances of unnormalized and normalized symmetric graph Laplacian matrices are similar, and they are better than that of normalized random walk graph Laplacian matrix. It is helpful to choose proper graph Laplacian matrix for PPI networks’ analysis.
Posted Content•
TL;DR: In this article, Nikiforov et al. showed that the maximum number of even cycles of length 1/4 can be computed for any positive integer ϵ > 0.
Abstract: As the counterpart of classical theorems on cycles of consecutive lengths due to Bondy and Bollob\'as in spectral graph theory, Nikiforov proposed the following open problem in 2008: What is the maximum $C$ such that for all positive $\varepsilon \sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$ for each integer $\ell\in[3,(C-\varepsilon)n]$. We prove that $C\geq\frac{1}{4}$ by a novel method, improving the existing bounds. Besides several novel ideas, our proof technique is partly inspirited by the recent research on Ramsey numbers of star versus large even cycles due to Allen, {\L}uczak, Polcyn and Zhang, and with aid of a powerful spectral inequality. We also derive an Erd\H{o}s-Gallai-type edge number condition for even cycles, which may be of independent interest.
TL;DR: An algorithm for computing the position and orientation of 3-D objects by comparing graphs by performing a spectral decomposition with obtaining eigenvectors of weighted adjacency matrix of the graph.
Abstract: This paper describes an algorithm for computing the position and orientation of 3-D objects by comparing graphs. The graphs are based on feature points of the image. Comparison is performed by a spectral decomposition with obtaining eigenvectors of weighted adjacency matrix of the graph.
TL;DR: Two algorithms for the evaluation of the characteristic polynomial of a graph G have the operation count of the order n^3, where n is the number of the vertices in the graph G and an algorithm requiring only O(n^2) operations is described.