Related papers: Unsolved Problems in Spectral Graph Theory
Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to…
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…
In this paper, we describe some recent spectral Moore theorems related to determining the maximum order of a connected graph of given valency and second eigenvalue. We show how these spectral Moore theorems have applications in Alon-Boppana…
We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of…
Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these…
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-Neumann operator. Over the past years, there has been…
The spectral theory of quantum graphs is related via an exact trace formula with the spectrum of the lengths of periodic orbits (cycles) on the graphs. The latter is a degenerate spectrum, and understanding its structure (i.e.,finding out…
Symplectic geometry plays an increasingly important role in mathematics, physics and applications, and naturally gives rise to interesting matrix families and properties. One of these is the notion of symplectic eigenvalues, whose existence…
This is a replacement paper. There are 6 chapters. The first two chapters are introductory. The third chapter is on extremal graph theory. The fourth chapter is about algebra in graph theory. The fifth chapter is focused on algorithms. The…
A complex unit gain graph is a graph where each orientation of an edge is given a complex unit, which is the inverse of the complex unit assigned to the opposite orientation. We extend some fundamental concepts from spectral graph theory to…
We present a general method for obtaining the spectra of large graphs with short cycles using ideas from statistical mechanics of disordered systems. This approach leads to an algorithm that determines the spectra of graphs up to a high…
The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks).…
A realization of a graph $G=(V,E)$ is a map $v\colon V\to\Bbb R^d$ that assigns to each vertex a point in $d$-dimensional Euclidean space. We study graph realizations from the perspective of representation theory (expressing certain…
Characterizing graphs by their spectra is a fundamental and challenging problem in spectral graph theory, which has received considerable attention in recent years. A major unsolved conjecture in this area is Haemers' conjecture which…
We define a graph to be $S$-regular if it contains an equitable partition given by a matrix $S$. These graphs are generalizations of both regular and bipartite, biregular graphs. An $S$-regular matrix is defined then as a matrix on an…
This paper develops analityc methods for investigating uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple setup is…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…
The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from non-commutative geometry. A set of axioms charaterising…
The inverse eigenvalue problem of a graph studies the real symmetric matrices whose off-diagonal pattern is prescribed by the adjacencies of the graph. The strong spectral property (SSP) is an important tool for this problem. This note…
This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications…