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 Analysis of (Sequences of) Graph Matrices

Abstract: We study the extreme singular values of incidence graph matrices, obtaining lower and upper estimates that are asymptotically tight. This analysis is then used for obtaining estimates on the spectral condition number of some weighted graph matrices. A short discussion on possible preconditioning strategies within interior-point methods for network flow problems is also included.

A class of unicyclic graphs determined by their Laplacian spectrum

Abstract: Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+ 4. 1. Introduction. Let G be a simple graph on n vertices and A(G) be its adja- cency matrix. Let dG(v) be the degree of vertex v in G, and D(G) be the diagonal matrix with the degrees of the corresponding vertices of G on the diagonal and zero elsewhere. Matrix Q(G) = D(G) − A(G) is called the Laplacian matrix of G. The eigenvalues of A(G) (resp., Q(G)) and the spectrum (which consists of eigenvalues) of A(G) (resp., Q(G)) are also called the adjacency (resp., Laplacian) eigenvalues of G and the adjacency (resp., Laplacian) spectrum of G. Since both matrices A(G) and Q(G) are real symmetric matrices, their eigenvalues are all real numbers. So we can assume that �1(G) � �2(G) � ��� � �n(G) and µ1(G) � µ2(G) � ��� � µn(G) = 0 are the adjacency eigenvalues and the Laplacian eigenvalues of G, respectively. Two graphs are adjacency (resp., Laplacian) cospectral if they have the same adjacency (resp., Laplacian) spectrum. Denote by �(G) = �(G;�) = det(�I − A(G)) and �(G;µ) = det(µI − Q(G)) the characteristic polynomial of adjacency matrix and Laplacian matrix of G, respectively. A graph is said to be determined by the adjacency (resp., Laplacian) spectrum if there is no non-isomorphic graph with the same adjacency (resp., Laplacian) spectrum. In general, the spectrum of a graph does not determine the graph and the question "Which graphs are determined by their spectrum?" ((3)) remains a difficult problem. For the background and some known results about this problem and related topics, we refer the readers to (4) and references therein. For the unicyclic graphs, Haemers

Infinite networks and variation of conductance functions in discrete Laplacians

Abstract: For a given infinite connected graph G = (V, E) and an arbitrary but fixed conductance function c, we study an associated graph Laplacian Δc; it is a generalized difference operator where the differences are measured across the edges E in G; and the conductance function c represents the corresponding coefficients. The graph Laplacian (a key tool in the study of infinite networks) acts in an energy Hilbert space ℋE computed from c. Using a certain Parseval frame, we study the spectral theoretic properties of graph Laplacians. In fact, for fixed c, there are two versions of the graph Laplacian, one defined naturally in the l2 space of V and the other in ℋE. The first is automatically selfadjoint, but the second involves a Krein extension. We prove that, as sets, the two spectra are the same, aside from the point 0. The point zero may be in the spectrum of the second, but not the first. We further study the fine structure of the respective spectra as the conductance function varies, showing now how the spect...

Energy gaps of Hamiltonians from graph Laplacians

Abstract: The Cheeger inequalities give an upper and lower bound on the spectral gap of discrete Laplacians defined on a graph in terms of the geometric characteristics of the graph We generalise this approach and we employ it to determine if a given discrete Hamiltonian with non-positive elements is gapped or not in the thermodynamic limit First, we define the graph that corresponds to such a generic Hamiltonian Then we present a suitable generalisation of the Cheeger inequalities that overcomes scaling deficiencies of the original version By employing simple examples we illustrate how the generalised Cheeger inequalities can successfully identify gapped or gapless phases and we comment on the computational complexity of this approach

Spectral Descriptors for Graph Matching

Abstract: In this paper, we consider the weighted graph matching problem. Recently, approaches to this problem based on spectral methods have gained significant attention. We propose two graph spectral descriptors based on the graph Laplacian, namely a Laplacian family signature (LFS) on nodes, and a pairwise heat kernel distance on edges. We show the stability of both our descriptors under small perturbation of edges and nodes. In addition, we show that our pairwise heat kernel distance is a noise-tolerant approximation of the classical adjacency matrix-based second order compatibility function. These nice properties suggest a descriptor-based matching scheme, for which we set up an integer quadratic problem (IQP) and apply an approximate solver to find a near optimal solution. We have tested our matching method on a set of randomly generated graphs, the widely-used CMU house sequence and a set of real images. These experiments show the superior performance of our selected node signatures and edge descriptors for graph matching, as compared with other existing signature-based matchings and adjacency matrix-based matchings.