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.

On the calculation of resonances by analytic continuation of eigenvalues from the stabilization graph.

Abstract: Resonances play a major role in a large variety of fields in physics and chemistry. Accordingly, there is a growing interest in methods designed to calculate them. Recently, Landau et al. proposed a new approach to analytically dilate a single eigenvalue from the stabilization graph into the complex plane. This approach, termed Resonances Via Pade (RVP), utilizes the Pade approximant and is based on a unique analysis of the stabilization graph. Yet, analytic continuation of eigenvalues from the stabilization graph into the complex plane is not a new idea. In 1975, Jordan suggested an analytic continuation method based on the branch point structure of the stabilization graph. The method was later modified by McCurdy and McNutt, and it is still being used today. We refer to this method as the Truncated Characteristic Polynomial (TCP) method. In this manuscript, we perform an in-depth comparison between the RVP and the TCP methods. We demonstrate that while both methods are important and complementary, the advantage of one method over the other is problem-dependent. Illustrative examples are provided in the manuscript.

Generalized Graph Matrix, Graph Geometry, Quantum Chemistry, and Optimal Description of Physicochemical Properties

Abstract: The generalized graph matrix Γ(x,v) (Estrada, E. Chem. Phys. Lett. 2001, 336, 247) is shown to encompass several of the applications of graph theory in physical chemistry in a more compact and effective way. It defines several n-Euclidean graph metrics, which simulate a graph defolding by changing the exponent v from 0 to 0.5 in a continuous way. This matrix is included in the formalism of the Huckel molecular orbital approach by considering that the resonance integrals between nonneighbor atoms are a function of the topological distance in terms of β. In doing so, the isospectrality between graphs disappears by changing the x parameter in this matrix as a consequence of considering the interactions between nonneighbor atoms. The Γ(x,v) matrix permits several of the “classical” topological indices to be (re)defined using only one graph invariant. These indices include the connectivity index, Balaban J index, Zagreb indices, Wiener index, and Harary indices, which are represented in an 8-dimensional space ...

Interactive segmentation of clustered cells via geodesic commute distance and constrained density weighted Nyström method

Abstract: An interactive method is proposed for complex cell segmentation, in particular of clustered cells This article has two main contributions: First, we explore a hybrid combination of the random walk and the geodesic graph based methods for image segmentation and propose the novel concept of geodesic commute distance to classify pixels The computation of geodesic commute distance requires an eigenvector decomposition of the weighted Laplacian matrix of a graph constructed from the image to be segmented Second, by incorporating pairwise constraints from seeds into the algorithm, we present a novel method for eigenvector decomposition, namely a constrained density weighted Nystrom method Both visual and quantitative comparison with other semiautomatic algorithms including Voronoi-based segmentation, grow cut, graph cuts, random walk, and geodesic method are given to evaluate the performance of the proposed method, which is a powerful tool for quantitative analysis of clustered cell images in live cell imaging (C) 2010 International Society for Advancement of Cytometry

An Optimization Approach to Locally-Biased Graph Algorithms

Abstract: Locally-biased graph algorithms are algorithms that attempt to find local or small-scale structure in a large data graph. In some cases, this can be accomplished by adding some sort of locality constraint and calling a traditional graph algorithm; but more interesting are locally-biased graph algorithms that compute answers by running a procedure that does not even look at most of the input graph. This corresponds more closely to what practitioners from various data science domains do, but it does not correspond well with the way that algorithmic and statistical theory is typically formulated. Recent work from several research communities has focused on developing locally-biased graph algorithms that come with strong complementary algorithmic and statistical theory and that are useful in practice in downstream data science applications. We provide a review and overview of this work, highlighting commonalities between seemingly different approaches, and highlighting promising directions for future work.

Some results on the Laplacian spectrum

Abstract: A graph is called a Laplacian integral graph if the spectrum of its Laplacian matrix consists of integers, and a graph G is said to be determined by its Laplacian spectrum if there does not exist other non-isomorphic graph H such that H and G share the same Laplacian spectrum. In this paper, we obtain a sharp upper bound for the algebraic connectivity of a graph, and identify all the Laplacian integral unicyclic, bicyclic graphs. Moreover, we show that all the Laplacian integral unicyclic, bicyclic graphs are determined by their Laplacian spectra.