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

Distance spectra of graphs: A survey

In 1971, Graham and Pollack established a relationship between the number of negative eigenvalues of the distance matrix and the addressing problem in data communication systems. They also proved that the determinant of the distance matrix of a tree is a function of the number of vertices only. Since then several mathematicians were interested in studying the spectral properties of the distance matrix of a connected graph. Computing the distance characteristic polynomial and its coefficients was the first research subject of interest. Thereafter, the eigenvalues attracted much more attention. In the present paper, we report on the results related to the distance matrix of a graph and its spectral properties.

Distance Spectra of Graphs: A Survey

An Introduction to the Theory of Graph Spectra: Introduction

Let G be a graph and denote by Q(G)=D(G)+A(G),L(G)=D(G)-A(G) the sum and the difference between the diagonal matrix of vertex degrees and the adjacency matrix of G,respectively. In this paper,some properties of the matrix Q(G)are studied. At the same time,anecessary and sufficient condition for the equality of the spectrum of Q(G) and L(G) is given.

Some properties of the spectrum of graphs

On the distance spectrum of graphs

Reliability evaluation of interconnection network is important to the design and maintenance of multiprocessor systems. The extra connectivity and the extra edge-connectivity are two important parameters for the reliability evaluation of interconnection networks. The ${\mbi {n}}$ -dimensional bijective connection network (in brief, BC network) includes several well known network models, such as, hypercubes, Mobius cubes, crossed cubes, and twisted cubes. In this paper, we explore the extra connectivity and the extra edge-connectivity of BC networks, and discuss the structure of BC networks with many faults. We obtain a sharp lower bound of ${{g}}$ -extra edge-connectivity of an ${\mbi {n}}$ -dimensional BC network for ${{n}} \geq 4$ and $1 \leq { {g}} \leq {2^{[{{{n}} \over 2}]}}$ . We also obtain a sharp lower bound of ${ {g}}$ -extra connectivity of an ${{n}}$ -dimensional BC network for ${{n}} \geq 4$ and $1 \leq { {g}} \leq 2{ {n}}$ which improves the result in [“Reliability evaluation of BC networks,” IEEE Trans. Computers, DOI: 10.1109/tc.2012.106.] for $1 \leq { {g}} \leq { {n}} - 3$ . Furthermore, we give a remark about exploring the ${ {g}}$ -extra edge-connectivity of BC networks for the more general ${\mbi {g}}$ , and we also characterize the structure of BC networks with many faulty nodes or links. As an application, we obtain several results on the ${\mbi {g}}$ -extra (edge-) connectivity and the structure of faulty networks on hypercubes, Mobius cubes, crossed cubes, and twisted cubes.

Huiqiu Lin

Papers

On the distance spectrum of graphs

Reliability Evaluation of BC Networks in Terms of the Extra Vertex- and Edge-Connectivity

Graphs determined by their Aα-spectra

A note on the Aα-spectral radius of graphs

On the Aα-spectral radius of a graph