Developments on spectral characterizations of graphs (original) (raw)
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Linear Algebra and its Applications, 2002
This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for 0 and −1 to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.
PROPERTIES OF GRAPHS IN RELATION TO THEIR SPECTRA
We will explain basic concepts of spectral graph theory, which studies graph properties via the algebraic characteristics of its adjacency matrix or Laplacian matrix. It turns out that investigating eigenvalues and eigenvectors of a graph provides unexpected and interesting results. Methods of spectral graph theory can be used to examine large random graphs and help tackle many difficult combinatorial problems (in particular because of recently invented algorithms which are able to compute eigenvalues and eigenvectors even of huge graphs in relatively short time). This approach uses highly efficient tools from advanced linear algebra, matrix theory and geometry. This paper will cover the following issues :
Which graphs are determined by their spectrum?
Linear Algebra and its Applications, 2003
For almost all graphs the answer to the question in the title is still unknown. Here we survey the cases for which the answer is known. Not only the adjacency matrix, but also other types of matrices, such as the Laplacian matrix, are considered.
The Laplacian spectrum of graphs
The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ 2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added.
Spectra of graphs and the spectral criterion for property (T)
Electronic Journal of Graph Theory and Applications
For a finite connected graph X, we consider the graph RX obtained from X by associating a new vertex to every edge of X and joining by edges the extremities of each edge of X to the corresponding new vertex. We express the spectrum of the Laplace operator on RX as a function of the corresponding spectrum on X. As a corollary, we show that X is a complete graph if and only if λ 1 (RX) > 1 2 . We give a re-interpretation of the correspondence X → RX in terms of the right-angled Coxeter group defined by X.
Properties and Recent Applications in Spectral Graph Theory
There are numerous applications of mathematics, specifically spectral graph theory, within the sciences and many other fields. This paper is an exploration of recent applications of spectral graph theory, including the fields of chemistry, biology, and graph coloring. Topics such as the isomers of alkanes, the importance of eigenvalues in protein structures, and the aid that the spectra of a graph provides when coloring a graph are covered, as well as others.
Some spectral properties of the non-backtracking matrix of a graph
Linear Algebra and its Applications, 2021
We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.
Universitext, 2012
Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices. Spectral graph theory is a useful subject. The founders of Google computed the Perron-Frobenius eigenvector of the web graph and became billionaires. The second largest eigenvalue of a graph gives information about expansion and randomness properties. The smallest eigenvalue gives information about independence number and chromatic number. Interlacing gives information about substructures. The fact that eigenvalue multiplicities must be integral provides strong restrictions. And the spectrum provides a useful invariant. This book gives the standard elementary material on spectra in Chapter 1. Important applications of graph spectra involve the largest or second largest or smallest eigenvalue, or interlacing, topics that are discussed in Chapters 3-4. Afterwards, special topics such as trees, groups and graphs, Euclidean representations, and strongly regular graphs are discussed. Strongly related to strongly regular graphs are regular two-graphs, and Chapter 10 mainly discusses Seidel's work on sets of equiangular lines. Strongly regular graphs form the first nontrivial case of (symmetric) association schemes, and Chapter 11 gives a very brief introduction to this topic, and Delsarte's Linear Programming Bound. Chapter 12 very briefly mentions the main facts on distance-regular graphs, including some major developments that occurred since the monograph [51] was written (proof of the Bannai-Ito conjecture, construction by Van Dam & Koolen of the twisted Grassmann graphs, determination of the connectivity of distance-regular graphs). Instead of working over R, one can work over F p or Z and obtain more detailed information. Chapter 13 considers pranks and Smith Normal Forms. Finally, Chapters 14 and 15 return to the real spectrum and consider in what cases a graph is determined by its spectrum, and when it has only few eigenvalues. v vi Preface Royle [169]. For association schemes and distance-regular graphs, see Bannai & Ito [19] and Brouwer, Cohen & Neumaier [51].
An Application of Hoffman Graphs for Spectral Characterizations of Graphs
The Electronic Journal of Combinatorics
In this paper, we present the first application of Hoffman graphs for spectral characterizations of graphs. In particular, we show that the 2-clique extension of the (t+1)times(t+1)(t+1)\times (t+1)(t+1)times(t+1)-grid is determined by its spectrum when ttt is large enough. This result will help to show that the Grassmann graph J_2(2D,D)J_2(2D,D)J_2(2D,D) is determined by its intersection numbers as a distance regular graph, if DDD is large enough.