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 perfect matching in a graph $G$ is a set of nonadjacent edges covering every vertex of $G$. Motivated by recent progress on the relations between the eigenvalues and the matching number of a graph, in this paper, we aim to present a distance spectral radius condition to guarantee the existence of a perfect matching. Let $G$ be an $n$-vertex connected graph where $n$ is even and $\lambda_{1}(D(G))$ be the distance spectral radius of $G$. Then the following statements are true. 
\noindent$\rm{I)}$ If $4\le n\le10$ and ${\lambda }_{1} (D\left(G\right))\le {\lambda }_{1} (D(S_{n,{\frac{n}{2}}-1}))$, then $G$ contains a perfect matching unless $G\cong S_{n,{\frac{n}{2}-1}}$ where $S_{n,{\frac{n}{2}-1}}\cong K_{{\frac{n}{2}-1}}\vee ({\frac{n}{2}+1})K_1$. 
\noindent$\rm{II)}$ If $n\ge 12$ and ${\lambda }_{1} (D\left(G\right))\le {\lambda }_{1} (D(G^*))$, then $G$ contains a perfect matching unless $G\cong G^*$ where $G^*\cong K_1\vee (K_{n-3}\cup2K_1)$. 
Moreover, if $G$ is a connected $2n$-vertex balanced bipartite graph with $\lambda_{1}(D(G))\le \lambda_{1}(D(B_{n-1,n-2})) $, then $G$ contains a perfect matching, unless $G\cong B_{n-1,n-2}$ where $B_{n-1,n-2}$ is obtained from $K_{n,n-2}$ by attaching two pendent vertices to a vertex in the $n$-vertex part.

Perfect matching and distance spectral radius in graphs and bipartite graphs

The spanning tree packing number of a graph $G$, denoted by $\tau(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $\tau(G)$ is one of the classic problems in graph theory. Cioab\u{a} and Wong initiated to investigate $\tau(G)$ from spectral perspectives in 2012 and since then, $\tau(G)$ has been well studied using the second largest eigenvalue of the adjacency matrix in the past decade. In this paper, we further extend the results in terms of the number of edges and the spectral radius, respectively; and prove tight sufficient conditions to guarantee $\tau(G)\geq k$ with extremal graphs characterized. Moreover, we confirm a conjecture of Ning, Lu and Wang on characterizing graphs with the maximum spectral radius among all graphs with a given order as well as fixed minimum degree and fixed edge connectivity. Our results have important applications in rigidity and nowhere-zero flows. We conclude with some open problems in the end.

Spectral radius and edge‐disjoint spanning trees

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $\rho(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $\rho(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $\rho(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $\rho(G)\geq \sqrt{m}$, then $G$ contains $K_{2,r+1}$ unless $G$ is a star; and $G$ contains either $C_3^+$ or $C_4^+$ unless $G$ is a complete bipartite graph, where $C_t^+$ denotes the graph obtained from $C_t$ and $C_3$ by identifying an edge. Secondly, we prove that if $\rho(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $\rho(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research.

Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs.

In this paper we prove that balanced characteristic functions of canonical cliques in a Paley graph of square order $P(q^2)$ span the $\frac{-1+q}{2}$-eigenspace of the graph. This is the first of two steps to a second proof of the analogue of Erd\"os-Ko-Rado theorem for Paley graphs of square order (the first proof was given by A. Blokhuis in 1984).

On balanced characteristic functions of canonical cliques in Paley graphs of square order

A $k$-tree is a spanning tree with every vertex of degree at most $k$. In this paper, we provide some sufficient conditions, in terms of the (adjacency and signless Laplacian) spectral radius, for the existence of a $k$-tree in a connected graph of order $n$, and a perfect matching in a balanced bipartite graph of order $2n$ with minimum degree $\delta$, respectively.

Huiqiu Lin

Papers

Perfect matching and distance spectral radius in graphs and bipartite graphs

Spectral radius and edge‐disjoint spanning trees

Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs.

On balanced characteristic functions of canonical cliques in Paley graphs of square order

The spanning $k$-trees, perfect matchings and spectral radius of graphs