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

Extremal spectral results related to spanning trees of signed complete graphs

Abstract: Signed graphs have their edges labeled either as positive or negative. ρ(M) denote the M-spectral radius of Σ, where M = M(Σ) is a real symmetric graph matrix of Σ. Obviously, ρ(M) = max{λ1(M),−λn(M)}. Let A(Σ) be the adjacency matrix of Σ and (Kn,H−) be a signed complete graph whose negative edges induce a subgraph H . In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum ρ(A(Σ)) among (Kn,T −) where T is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum λ1(A(Σ)) and minimum λn(A(Σ)) among (Kn,T −), respectively. Another interesting graph matrix of a signed graph is distance matrix, i.e. D(Σ)which was defined by Hameed, Shijin, Soorya, Germina and Zaslavsky [8]. Note that A(Σ) = D(Σ) when Σ ∈ (Kn,T−). In this paper, we give upper bounds on the least distance eigenvalue of a signed graph Σ with diameter at least 2. This result implies a result proved by Lin [11] was originally conjectured by Aouchiche and Hansen [1].

Eigenvalues of signed graphs

Given a planar graph family $\mathcal{F}$, let ${\rm ex}_{\mathcal{P}}(n,\mathcal{F})$ and ${\rm spex}_{\mathcal{P}}(n,\mathcal{F})$ be the maximum size and maximum spectral radius over all $n$-vertex $\mathcal{F}$-free planar graphs, respectively. Let $tC_k$ be the disjoint union of $t$ copies of $k$-cycles, and $t\mathcal{C}$ be the family of $t$ vertex-disjoint cycles without length restriction. Tait and Tobin [Three conjectures in extremal spectral graph theory, J. Combin. Theory Ser. B 126 (2017) 137--161] determined that $K_2+P_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order $n$, which implies the extreme graphs of $spex_{\mathcal{P}}(n,tC_{\ell})$ and $spex_{\mathcal{P}}(n,t\mathcal{C})$ for $t\geq 3$ are $K_2+P_{n-2}$. In this paper, we first determine $spex_{\mathcal{P}}(n,tC_{\ell})$ and $spex_{\mathcal{P}}(n,t\mathcal{C})$ and characterize the unique extremal graph for $1\leq t\leq 2$, $\ell\geq 3$ and sufficiently large $n$. Secondly, we obtain the exact values of ${\rm ex}_{\mathcal{P}}(n,2C_4)$ and ${\rm ex}_{\mathcal{P}}(n,2\mathcal{C})$, which answers a conjecture of Li [Planar Tur\'an number of disjoint union of $C_3$ and $C_4$, arxiv:2212.12751v1 (2022)]. These present a new exploration of approaches and tools to investigate extremal problems of planar graphs.

Extremal spectral results of planar graphs without vertex-disjoint cycles

Corrigendum to “Colorings and spectral radius of digraphs” [Discrete Math. 339 (1) (2016) 327–332]

Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal graphs. This result gives spectral conditions for the existence of specified paths and cycles in graphs.

Huiqiu Lin

Papers

Extremal spectral results related to spanning trees of signed complete graphs

Eigenvalues of signed graphs

Extremal spectral results of planar graphs without vertex-disjoint cycles

Corrigendum to “Colorings and spectral radius of digraphs” [Discrete Math. 339 (1) (2016) 327–332]

Spectral conditions for the existence of specified paths and cycles in graphs