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

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities are called the \emph{$A_{\alpha}$-spectrum} of $G$. A graph $G$ is said to be \emph{determined by its $A_{\alpha}$-spectrum} if all graphs having the same $A_{\alpha}$-spectrum as $G$ are isomorphic to $G$. We first prove that some graphs are determined by its $A_{\alpha}$-spectrum for $0\leq\alpha<1$, including the complete graph $K_m$, the star $K_{1,n-1}$, the path $P_n$, the union of cycles and the complement of the union of cycles, the union of $K_2$ and $K_1$ and the complement of the union of $K_2$ and $K_1$, and the complement of $P_n$. Setting $\alpha=0$ or $\frac{1}{2}$, those graphs are determined by $A$- or $Q$-spectra. Secondly, when $G$ is regular, we show that $G$ is determined by its $A_{\alpha}$-spectrum if and only if the join $G\vee K_m$ is determined by its $A_{\alpha}$-spectrum for $\frac{1}{2}<\alpha<1$. Furthermore, we also show that the join $K_m\vee P_n$ is determined by its $A_{\alpha}$-spectrum for $\frac{1}{2}<\alpha<1$. In the end, we pose some related open problems for future study.

Graphs determined by their $A_{\alpha}$-spectra

On cyclic edge connectivity of regular graphs with two orbits.

Distance spectrum, 1-factor and vertex-disjoint cycles

Rigidity is the property of a structure that does not flex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characterization of rigid graphs in $\mathbb{R}^2$. Lovász and Yemini (1982) proved that every $6$-connected graph is rigid in $\mathbb{R}^2$. Jackson and Jordán (2005) strengthened this result, and showed that every $6$-connected graph is globally rigid in $\mathbb{R}^2$. Thus every graph with algebraic connectivity greater than $5$ is globally rigid in $\mathbb{R}^2$. In 2021, Cioabă, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least $6$ and algebraic connectivity greater than $2+\frac{1}{\delta-1}$ (resp., $2+\frac{2}{\delta-1}$) is rigid (resp., globally rigid) in $\mathbb{R}^2$. In this paper, we study the rigidity of graphs in $\mathbb{R}^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius condition for the rigidity (resp., globally rigidity) of $2$-connected (resp., $3$-connected) graphs with given minimum degree. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order $n$.

/pdf/spectral-radius-conditions-for-the-rigidity-of-graphs-1tqfhj6v.pdf

Huiqiu Lin

Papers

Graphs determined by their $A_{\alpha}$-spectra

On cyclic edge connectivity of regular graphs with two orbits.

Distance spectrum, 1-factor and vertex-disjoint cycles

Spectral Radius Conditions for the Rigidity of Graphs

Corrigendum to “A note on (signless) Laplacian spectral ordering with maximum degrees of graphs” [Linear Algebra Appl. 521 (2017) 135–141]