Spectral Radius Of Graphs

Spectral Radius of Graphs provides a thorough overview of important results on the spectral radius of adjacency matrix of graphs that have appeared in the literature in the preceding ten years, most of them with proofs, and including some previously unpublished results of the author. The primer begins with a brief classical review, in order to provide the reader with a foundation for the subsequent chapters. Topics covered include spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem. From this introduction, the book delves deeper into the properties of the principal eigenvector; a critical subject as many of the results on the spectral radius of graphs rely on the properties of the principal eigenvector for their proofs. A following chapter surveys spectral radius of special graphs, covering multipartite graphs, non-regular graphs, planar graphs, threshold graphs, and others. Finally, the work explores results on the structure of graphs having extreme spectral radius in classes of graphs defined by fixing the value of a particular, integer-valued graph invariant, such as: the diameter, the radius, the domination number, the matching number, the clique number, the independence number, the chromatic number or the sequence of vertex degrees. Throughout, the text includes the valuable addition of proofs to accompany the majority of presented results. This enables the reader to learn tricks of the trade and easily see if some of the techniques apply to a current research problem, without having to spend time on searching for the original articles. The book also contains a handful of open problems on the topic that might provide initiative for the reader's research. Dedicated coverage to one of the most prominent graph eigenvalues Proofs and open problems included for further study Overview of classical topics such as spectral decomposition, the Perron-Frobenius theorem, the Rayleigh quotient, the Weyl inequalities, and the Interlacing theorem

This book explores the inequalities for eigenvalues of the six matrices associated with graphs. Includes the main results and selected applications.

We determine a lower bound for the spectral radius of a graph in terms of the number of vertices and the diameter of the graph. For the specific case of graphs with diameter three we give a slightly better bound. We also construct families of graphs with small spectral radius, thus obtaining asymptotic results showing that the bound is of the right order. We also relate these results to the extremal degree/diameter problem.

Graphs and matrices enjoy a fascinating and mutually beneficial relationship. This interplay has benefited both graph theory and linear algebra. In one direction, knowledge about one of the graphs that can be associated with a matrix can be used to illuminate matrix properties and to get better information about the matrix. Examples include the use of digraphs to obtain strong results on diagonal dominance and eigenvalue inclusion regions and the use of the Rado-Hall theorem to deduce properties of special classes of matrices. Going the other way, linear algebraic properties of one of the matrices associated with a graph can be used to obtain useful combinatorial information about the graph. The adjacency matrix and the Laplacian matrix are two well-known matrices associated to a graph, and their eigenvalues encode important information about the graph. Another important linear algebraic invariant associated with a graph is the Colin de Verdiere number, which, for instance, characterizes certain topological properties of the graph. This book is not a comprehensive study of graphs and matrices. The particular content of the lectures was chosen for its accessibility, beauty, and current relevance, and for the possibility of enticing the audience to want to learn more.

The adjacency matrix of a graph is a (0, 1)-matrix indexed by the vertex set of the graph. And the signless Laplacian matrix of a graph is the sum of its adjacency matrix and its diagonal matrix of vertex degrees. The eigenvalues and the signless Laplacian eigenvalues of a graph are the eigenvalues of the adjacency matrix and the signless Laplacian matrix, respectively. These two matrices of a graph have been studied for several decades since they have been applied to many research field, such as computer science, communication network, information science and so on. In this thesis, we study k-connected graphs and focus on their spectral radius and signless Laplacian spectral radius. Firstly, we determine the graphs with maximum spectral radius among all k-connected graphs of fixed order with given diameter. As we know, when a graph is regular, its spectral radius and signless Laplacian spectral radius can easily be found. We obtain an upper bound on the signless Laplacian spectral radius of k-connected irregular graphs. Finally, we give some other results mainly related to the signless Laplacian matrix.