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.

Spectral Graph Cut from a Filtering Point of View

Abstract: Spectral graph theory is well known and widely used in computer vision. In this paper, we analyze image segmentation algorithms that are based on spectral graph theory, e.g., normalized cut, and show that there is a natural connection between spectural graph theory based image segmentationand and edge preserving filtering. Based on this connection we show that the normalized cut algorithm is equivalent to repeated iterations of bilateral filtering. Then, using this equivalence we present and implement a fast normalized cut algorithm for image segmentation. Experiments show that our implementation can solve the original optimization problem in the normalized cut algorithm 10 to 100 times faster. Furthermore, we present a new algorithm called conditioned normalized cut for image segmentation that can easily incorporate color image patches and demonstrate how this segmentation problem can be solved with edge preserving filtering.

On the Normalized Laplacian Spectrum of Graphs and Graph Operations

Abstract: For any graph Γ, we can define several matrices using the connection between its vertices. Evaluation of structural information of a graph from the eigenvalues of a matrix associated to it is the main goal of spectral graph theory. The adjacency matrix, Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix are the most popular matrices studied in spectral graph theory. In this thesis, our main focus is to analyze the eigenvalues of the normalized Laplacian matrix of a graph. For any undirected graph Γ (without any isolated vertices), let A be the (0,1)-adjacency matrix of Γ. The normalized Laplacian of Γ is Δ = I −D−1A, where D is the diagonal matrix of vertex degrees of Γ. If Γ is connected, then the matrix A = D⁻¹A is an irreducible row-stochastic matrix. The previously existing eigenvalue localization theorems did not provide satisfactory eigenvalue bounds for A. Here we come up with a new eigenvalue localization theorem, by applying which we get improved bounds for the eigenvalues of A as well as of Δ. The Randi´c index of a graph is useful to characterize that graph. It is also an useful tool to bound the normalized Laplacian energy of a graph. We generalize the concept of the Randi´c index and introduce some new topological indices. We call them general Randi´c indices for matching. We show that the coefficients of the characteristic polynomial, of the normalized Laplacian, of a tree can be expressed by these indices. Finally, we characterize these results for two special class of trees. It is not always easy to find the spectrum of a graph with large number of vertices and it is also seen that many real-world networks possess special type of graph structure. Creation of those structures can be explained by some graph operations, namely, vertex doubling, motif (induced subgraph) doubling, motif joining, etc. These graph operations produce certain eigenvalues, like 1, 1±0.5, 1±√0.5 etc, which are mostly observed in the normalized graph Laplacian of many real networks. For example, the doubling of a vertex always ensures the eigenvalue 1. We investigate the emergence of particular eigenvalues, such as, eigenvalue 1 and others by the above-mentioned graph operations. A threshold graph is an iterated graph. Production of a threshold graph can also be considered as a sequence of graph operations, starting from a single vertex, by repeatedly performing one of the two graph operations, namely, (a) addition of a single isolated vertex to the graph or (b) addition of a single dominating vertex to the graph. Thus a threshold graph can always be represented by a unique binary string starting with 0. We show that the unique string provides certain eigenvalues of that graph. Finally, we try to characterize threshold graphs with few distinct eigenvalues.

Community Detection by $L_0$-penalized Graph Laplacian

Abstract: Community detection in network analysis aims at partitioning nodes in a network into $K$ disjoint communities. Most currently available algorithms assume that $K$ is known, but choosing a correct $K$ is generally very difficult for real networks. In addition, many real networks contain outlier nodes not belonging to any community, but currently very few algorithm can handle networks with outliers. In this paper, we propose a novel model free tightness criterion and an efficient algorithm to maximize this criterion for community detection. This tightness criterion is closely related with the graph Laplacian with $L_0$ penalty. Unlike most community detection methods, our method does not require a known $K$ and can properly detect communities in networks with outliers. Both theoretical and numerical properties of the method are analyzed. The theoretical result guarantees that, under the degree corrected stochastic block model, even for networks with outliers, the maximizer of the tightness criterion can extract communities with small misclassification rates even when the number of communities grows to infinity as the network size grows. Simulation study shows that the proposed method can recover true communities more accurately than other methods. Applications to a college football data and a yeast protein-protein interaction data also reveal that the proposed method performs significantly better.

Bounding robustness in complex networks under topological changes through majorization techniques

Abstract: Measuring robustness is a fundamental task for analysing the structure of complex networks. Indeed, several approaches to capture the robustness properties of a network have been proposed. In this paper we focus on spectral graph theory where robustness is measured by means of a graph invariant called Kirchhoff index, expressed in terms of eigenvalues of the Laplacian matrix associated to a graph. This graph metric is highly informative as a robustness indicator for several real-world networks that can be modeled as graphs. We discuss a methodology aimed at obtaining some new and tighter bounds of this graph invariant when links are added or removed. We take advantage of real analysis techniques, based on majorization theory and optimization of functions which preserve the majorization order. Applications to simulated graphs and to empirical networks generated by collecting assets of the S&P 100 show the effectiveness of our bounds, also in providing meaningful insights with respect to the results obtained in the literature.

On the (Laplacian) spectral radius of unicyclic graphs with girth g and k pendant vertices

Abstract: The spectral radius of a graph is the largest eigenvalue of adjacency matrix of the graph and its Laplacian spectral radius is the largest eigenvalue of the Laplacian matrix which is the difference of the diagonal matrix of vertex degrees and the adjacency matrix In this paper, we determine the unicyclic graph with the maximal spectral radius and the maximal Laplacian spectral radius among all unicyclic graphs of order n and girth g with k pendant vertices, respectively