Let G be a simple graph with V (E) as its vertex (respectively, edge) set. The spectrum of G is the spectrum of its adjacency matrix. For all other definitions (or notation not given here), especially those related to graph spectra, see [1]. The purpose of this note is to provide shorter proofs of two inequalities already known in spectral graph theory. The first bounds vertex eccentricities of...

We demonstrate how Monte Carlo Search (MCS) algorithms, namely Nested (NMCS) and Rollout Policy Adaptation (NRPA), can be used to build graphs find counter-examples spectral graph theory conjectures in minutes.

We will introduce spectral graph theory by seeing the value of studying the eigenvalues of various matrices associated with a graph. Then, we will learn about applications to the study of expanders and Ramanujan graphs, and more generally, to computer science as a whole.

Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3.

With every graph (or digraph) one can associate several different matrices. Here we shall concentrate mainly on the adjacency matrix of (undirected) graphs, and also discuss briefly the Laplacian. We shall show that spectral properties (the eigenvalues and eigenvectors) of these matrices provide useful information about the structure of the graph. It turns out that for regular graphs, the infor...

We present a spectral theory of uniform hypergraphs that closely parallels Spectral Graph Theory. A number of recent developments building upon classical work has led to a rich understanding of “symmetric hyperdeterminants” of hypermatrices, a.k.a. multidimensional arrays. Symmetric hyperdeterminants share many properties with determinants, but the context of multilinear algebra is substantiall...

Let graph energy is a graph--spectrum--based quantity‎, ‎introduced in the 1970s‎. ‎After a latent period of 20--30 years‎, ‎it became a popular topic of research both‎ ‎in mathematical chemistry and in ``pure'' spectral graph theory‎, ‎resulting in‎ ‎over 600 published papers‎. ‎Eventually‎, ‎scores of different graph energies have‎ ‎been conceived‎. ‎In this article we...

In 2003 Vladimir Nikiforov [7] began a line of research whose aim was to build an extremal theory of graphs based on spectral theory. We will discuss some of his results and in particular we will focus on a result of Babai and Guiduli [1] that gives a Kövari-Sós-Turán type upper bound on the largest eigenvalue of the adjacency matrix of a Ks,t-free graph. This spectral approach sheds new light ...

Algebraic combinatorics is the area of mathematics that uses the theories and methods of abstract and linear algebra to solve combinatorial problems, or conversely applies combinatorial techniques to solve problems in algebra. In particular, spectral graph theory applies the techniques of linear algebra to study graph theory. Spectral graph theory is the study of the eigenvalues of various matr...