Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

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An Introduction To The Theory Of Graph Spectra

In 1962, Erdős gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number and minimum degree of graphs which generalized Ore’s theorem. One year later, Moon and Moser gave an analogous result for Hamilton cycles in balanced bipartite graphs. In this paper, we present the spectral analogues of Erdős’ theorem and Moon–Moser’s theorem, respectively. Let be the class of non-Hamiltonian graphs of order n and minimum degree at least k. We determine the maximum (signless Laplacian) spectral radius of graphs in (for large enough n), and the minimum (signless Laplacian) spectral radius of the complements of graphs in . All extremal graphs with the maximum (signless Laplacian) spectral radius and with the minimum (signless Laplacian) spectral radius of the complements are determined, respectively. We also solve similar problems for balanced bipartite graphs and the quasi-complements.

Spectral analogues of Erdős’ and Moon–Moser’s theorems on Hamilton cycles

Written for mathematicians working with the theory of graph spectra, this book explores more than 400 inequalities for eigenvalues of the six matrices associated with finite simple graphs: the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, Seidel matrix, and distance matrix. The book begins with a brief survey of the main results and selected applications to related topics, including chemistry, physics, biology, computer science, and control theory. The author then proceeds to detail proofs, discussions, comparisons, examples, and exercises. Each chapter ends with a brief survey of further results. The author also points to open problems and gives ideas for further reading.

Inequalities for Graph Eigenvalues

Let G be a graph of order n and λ(G) the spectral radius of its adjacency matrix. We extend some recent results on sufficient conditions for Hamiltonian paths and cycles in G. One of the main results of the paper is the following theorem

Spectral radius and Hamiltonicity of graphs with large minimum degree

Sufficient spectral conditions on Hamiltonian and traceable graphs

In this paper, we give sufficient conditions on the spectral radius for a bipartite graph to Hamiltonian and traceable, which expand the results of Lu, Liu and Tian (2012) [10]. Furthermore, we also present tight sufficient conditions on the signless Laplacian spectral radius for a graph to Hamiltonian and traceable, which improve the results of Yu and Fan (2012) [12].

Ruifang Liu

Papers

Sufficient spectral conditions on Hamiltonian and traceable graphs

Sufficient spectral conditions on Hamiltonian and traceable graphs

The least eigenvalue of unicyclic graphs with n vertices and k pendant vertices

The inertia and energy of the distance matrix of a connected graph

On the spectral radius of bipartite graphs with given diameter