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

Over the past half century, the rigidity of graphs in R 2 has aroused a great deal of interest. Lov´asz and Yemini (1982) proved that every 6-connected graph is rigid in R 2 . Jackson and Jord´an (2005) provided a similar vertex-connectivity condition for the globally rigidity of graphs in R 2 . These results imply that a graph G with algebraic connectivity µ ( G ) > 5 is (globally) rigid in R 2 . Cioab˘a, Dewar and Gu (2021) improved this bound, and proved that a graph G with minimum degree δ ≥ 6 is rigid in R 2 if µ ( G ) > 2 + 1 δ − 1 , and is globally rigid in R 2 if µ ( G ) > 2 + 2 δ − 1 . In this paper, we study the (globally) rigidity of graphs in R 2 from the viewpoint of adjacency eigenvalues. Speciﬁcally, we provide suﬃcient conditions for a 2-connected (resp. 3-connected) graph with given minimum degree to be rigid (resp. globally rigid) in terms of the spectral radius. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order n .

Spectral radius and (globally) rigidity of graphs in $R^2$

The signed graphs with all but at most three eigenvalues equal to −1

A mixed circulant graph is called integral if all eigenvalues of its Hermitian adjacency matrix are integers. The main purpose of this paper is to investigate the existence of perfect state transfer (PST for short) and multiple state transfer (MST for short) on integral mixed circulant graphs. Concretely, we provide suﬃcient and necessary conditions for the existence of PST and MST between speciﬁed pairs of vertices on integral mixed circulant graphs, respectively.

State transfer on integral mixed circulant graphs

Let $G$ be a connected graph of order $n$ with diameter $d$. Remoteness $\rho$ of $G$ is the maximum average distance from a vertex to all others and $\partial_1\geq\cdots\geq \partial_n$ are the distance eigenvalues of $G$. In \cite{AH}, Aouchiche and Hansen conjectured that $\rho+\partial_3>0$ when $d\geq 3$ and $\rho+\partial_{\lfloor\frac{7d}{8}\rfloor}>0.$ In this paper, we confirm these two conjectures. Furthermore, we give lower bounds on $\partial_n+\rho$ and $\partial_1-\rho$ when $G\ncong K_n$ and the extremal graphs are characterized.

Remoteness and distance eigenvalues of a graph

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Huiqiu Lin

Papers

Spectral radius and (globally) rigidity of graphs in $R^2$

The signed graphs with all but at most three eigenvalues equal to −1

State transfer on integral mixed circulant graphs

Remoteness and distance eigenvalues of a graph

Toughness, hamiltonicity and spectral radius in graphs