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

A spectral condition for odd cycles in non-bipartite graphs

Perfect matching and distance spectral radius in graphs and bipartite graphs

On the sum of k largest distance eigenvalues of graphs

Let be a connected graph with vertex set and edge set E(G). Let D(G) be the distance matrix of G and be its distance spectrum. The distance between distance spectra of G and is defined byDefine the cospectrality of G byLet . In the paper, we obtain lower bounds on and for . Furthermore, we give an upper bound on .

Distance between distance spectra of graphs

Let $G$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$ Let $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$ and $D(G)$ be the distance matrix of $G$ The distance Laplacian matrix of $G$ is defined as $\mathcal{L}(G)=Tr(G)-D(G)$ The distance signless Laplacian matrix of $G$ is defined as $\mathcal{Q}(G)=Tr(G)+D(G)$ In this paper, we give a lower bound on the distance Laplacian spectral radius in terms of $D_1$, as a consequence, we show that $\partial_1^L(G)\geq n+\lceil\frac{n}{\omega}\rceil$ where $\omega$ is the clique number of $G$ Furthermore, we give some graft transformations, by using them, we characterize the extremal graph attains the maximum distance spectral radius in terms of $n$ and $\omega$ Moreover, we also give bounds on the distance signless Laplacian eigenvalues of $G$, and give a confirmation on a conjecture due to Aouchiche and Hansen

Huiqiu Lin

Papers

A spectral condition for odd cycles in non-bipartite graphs

Perfect matching and distance spectral radius in graphs and bipartite graphs

On the sum of k largest distance eigenvalues of graphs

Distance between distance spectra of graphs

More results on the distance (signless) Laplacian eigenvalues of graphs