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.

Discovering Genetic Ancestry Using Spectral Graph Theory

Abstract: As one approach to uncovering the genetic underpinnings of complex disease, individuals are measured at a large number of genetic variants (usually SNPs) across the genome and these SNP genotypes are assessed for association with disease status. We propose a new statistical method called Spectral-GEM for the analysis of genome-wide association studies; the goal of Spectral-GEM is to quantify the ancestry of the sample from such genotypic data. Ignoring structure due to differential ancestry can lead to an excess of spurious findings and reduce power. Ancestry is commonly estimated using the eigenvectors derived from principal component analysis (PCA). To develop an alternative to PCA we draw on connections between multidimensional scaling and spectral graph theory. Our approach, based on a spectral embedding derived from the normalized Laplacian of a graph, can produce more meaningful delineation of ancestry than by using PCA. Often the results from Spectral-GEM are straightforward to interpret and therefore useful in association analysis. We illustrate the new algorithm with an analysis of the POPRES data [Nelson et al., 2008].

Intra-Retinal Layer Segmentation of 3D Optical Coherence Tomography Using Coarse Grained Diffusion Map

Abstract: Optical coherence tomography (OCT) is a powerful and noninvasive method for retinal imaging. In this paper, we introduce a fast segmentation method based on a new variant of spectral graph theory named diffusion maps. The research is performed on spectral domain (SD) OCT images depicting macular and optic nerve head appearance. The presented approach does not require edge-based image information and relies on regional image texture. Consequently, the proposed method demonstrates robustness in situations of low image contrast or poor layer-to-layer image gradients. Diffusion mapping is applied to 2D and 3D OCT datasets composed of two steps, one for partitioning the data into important and less important sections, and another one for localization of internal this http URL the first step, the pixels/voxels are grouped in rectangular/cubic sets to form a graph node.The weights of a graph are calculated based on geometric distances between pixels/voxels and differences of their mean intensity.The first diffusion map clusters the data into three parts, the second of which is the area of interest. The other two sections are eliminated from the remaining calculations. In the second step, the remaining area is subjected to another diffusion map assessment and the internal layers are localized based on their textural similarities.The proposed method was tested on 23 datasets from two patient groups (glaucoma and normals). The mean unsigned border positioning errors(mean - SD) was 8.52 - 3.13 and 7.56 - 2.95 micrometer for the 2D and 3D methods, respectively.

Atomic branching in molecules

Abstract: A graph theoretic measure of extended atomic branching is defined that accounts for the effects of all atoms in the molecule, giving higher weight to the nearest neighbors. It is based on the counting of all substructures in which an atom takes part in a molecule. We prove a theorem that permits the exact calculation of this measure based on the eigenvalues and eigenvectors of the adjacency matrix of the graph representing a molecule. The definition of this measure within the context of the Huckel molecular orbital (HMO) and its calculation for benzenoid hydrocarbons are also studied. We show that the extended atomic branching can be defined using any real symmetric matrix, as well as any Hermitian (self-adjoint) matrix, which permits its calculation in topological, geometrical, and quantum chemical contexts. © 2005 Wiley