The smallest eigenvalue of K_p-free graphs

Recent progress on graphs with fixed smallest eigenvalue

We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed…

On the largest two and smallest six distance Pareto eigenvalues of a graph

In this article, we establish some bounds involving the largest two distance Pareto eigenvalues of a connected graph. Also we characterize all possible values for smallest six distance Pareto eigenvalues of a connected graph.

An inequality involving the second largest and smallest eigenvalue of a distance-regular graph

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study distance-regular graphs with fixed second largest eigenvalue.

直径至少为 $3$ 且最小特征值至少为 $-3$ 的非几何距离正则图

Non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$

本文分类了直径至少为 $3$ 且最小特征值至少为 $-3$ 的非几何距离正则图，为最终完整分类该类图迈出了关键一步。

关于图的最大与最小特征值乘积的注记

Note on the product of the largest and the smallest eigenvalue of a graph

本文利用特征值交错推导出图的最大度与其最大、最小邻接特征值之间的不等式，并完全刻画了等号成立的情形。

Characterizing the second smallest eigenvalue of the normalized Laplacian of a tree

In this paper we show a monotonicity theorem for the harmonic eigenfunction of λ_{1} of the normalized Laplacian over the points of articulation of a graph. We introduce the definition of Perron component for the normalized Laplacian matrix of a graph and show how its second smallest eigenvalue can be characterized using this definition.

On the Smallest Eigenvalue of Grounded Laplacian Matrices

We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show…

非二部连通图的最小规范化无符号∞-拉普拉斯特征值

The smallest normalized signless $\infty$-Laplacian eigenvalue for non-bipartite connected graphs

本文研究了非二部连通图的最小规范化无符号∞-拉普拉斯特征值，证明其等于加权无符号关联矩阵广义逆的最小∞-范数的倒数，并举例说明。

具有固定最小特征值的正则图的结构理论

A structure theory for regular graphs with fixed smallest eigenvalue

本文建立固定最小特征值正则图的结构理论，并据此证明最小特征值为 -λ 的 k-正则图在 k 相对 λ 足够大时，其团数关于 k 线性增长。

Mixed graphs with smallest eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$

The classical problem of characterizing the graphs with bounded eigenvalues may date back to the work of Smith in 1970. Especially, the research on graphs with smallest eigenvalues not less than $-2$ has attracted widespread attention. Mixed graphs are natural generalization of undirected graphs. In this paper, we completely characterize the mixed graphs with smallest Hermitian eigenvalue greater than $-\frac{\sqrt{5}+1}{2}$, which consists of three infinite classes of mixed graphs and $30$ scattered mixed graphs. By the way, we get a new class of mixed graphs switching equivalent to their und…

On Graphs with the Smallest Eigenvalue at Least $-1-\sqrt{2}$, part II

This is a continuation of the article with the same title. In this paper, the family H is the same as in the previous paper "On Graphs with the Smallest Eigenvalue at Least $-1-\sqrt{2}$, part I". The main result is that a minimal graph which is not an H -line graph, is just isomorphic to one of the 38 graphs found by computer.

The unicyclic graphs with the second smallest normalized Laplacian eigenvalue no less than $1-\frac{\sqrt{6}}{3}$

Let $λ_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\geq21$ with $λ_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs with $λ_{2}(G)=1-\frac{\sqrt{6}}{3}$ are also determined.

Sesqui-regular graphs with fixed smallest eigenvalue

Let $λ\geq2$ be an integer. For strongly regular graphs with parameters $(v, k, a,c)$ and smallest eigenvalue $-λ$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. In this paper, we will study a new class of regular graphs called sesqui-regular graphs, which contains strongly regular graphs as a subclass, and prove that for a sesqui-regular graph with parameters $(v,k,c)$ and smallest eigenvalue at least $-λ$, if $k$ is very large, then either $c \leq λ^2(λ-1)$ or $v-k-1 \leq \frac{(λ-1)^2}{4} + 1$ holds.

汉明图及其推广的最小特征值和量子色数

On the Smallest Eigenvalues and Quantum Chromatic Numbers of Hamming Graphs and Generalizations

本文研究了距离参数小于一半长度的汉明图的最小特征值，给出了渐近下界，并应用于量子色数的下界。

On the second smallest and the largest normalized Laplacian eigenvalues of a graph

Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$. Let $λ_{k}(G)$ be the $k$-th smallest normalized Laplacian eigenvalue of $G$. Denote $ρ(A)$ the spectral radius of the matrix $A$. In this paper, we study the behaviors of $λ_{2}(G)$ and $ρ(\mathcal{L}(G))$ when the graph is perturbed by three operations.

Signed analogue of line graphs and their smallest eigenvalues

In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The proof is based on what we call Hoffman's limit theorem which we formulate for Hermitian matrices, and also the extension of the concept of Hoffman graph and line graph for the setting of signed graphs.

Max k-cut and the smallest eigenvalue

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right) $ be the maximum size of a $k$-cut of $G.$ It is shown that \[ \mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{μ_{\min }\left( G\right) n}{2}\right) , \] where $μ_{\min}\left( G\right) $ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.

强正则图最小特征值为$-m$时的改进界

An improved bound for strongly regular graphs with smallest eigenvalue $-m$

本文改进了强正则图在最小特征值为$-m$时的界，并展示了方法在推导正交数组Bruck完成定理中的应用。

On fat Hoffman graphs with smallest eigenvalue at least -3

We investigate fat Hoffman graphs with smallest eigenvalue at least -3, using their special graphs. We show that the special graph S(H) of an indecomposable fat Hoffman graph H is represented by the standard lattice or an irreducible root lattice. Moreover, we show that if the special graph admits an integral representation, that is, the lattice spanned by it is not an exceptional root lattice, then the special graph S(H) is isomorphic to one of the Dynkin graphs A_n, D_n, or extended Dynkin graphs A_n or D_n.

最小特征值至少为$-3$的半正则图的可积性

The integrability of sesqui-regular graphs with smallest eigenvalue at least $-3$

本文证明参数满足$c\geq3$且最小特征值在$[-3,-2)$的连通半正则图在度数足够大时是$1$-可积的，并指出$c$的下界最优，通过Shrikhande图与$K_t$的笛卡尔积给出反例。

具有固定最小特征值的有向图结构理论

A structure theory for signed graphs with fixed smallest eigenvalue

本文研究了具有固定最小特征值的有向图结构，并证明了在特定条件下，此类图具有密集子图和特定的特征值下界性质。

The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes.

矩阵的最大与最小特征值及图的若干哈密顿性质

The Largest and Smallest Eigenvalues of Matrices and Some Hamiltonian Properties of Graphs

本文利用矩阵 $αD+βA$ 在 $α\geq β>0$ 时的最大与最小特征值，建立了图具有哈密顿性和可迹性的充分条件。

The clique number and the smallest Q-eigenvalue of graphs

Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order $n,$ with no complete subgraph of order $r+1?$ It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on $q_{\min}$ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum $q_{\m…

Note on the sum of the smallest and largest eigenvalues of a triangle-free graph

Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrix eigenvalues $μ_1(G)\geq μ_2(G)\geq \dots \geq μ_n(G)$. In this paper we study the quantity $$μ_1(G)+μ_n(G).$$ We prove that for any triangle-free graph $G$ we have $$μ_1(G)+μ_n(G)\leq (3-2\sqrt{2})n.$$ This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum of $$\frac{μ_1(G)+μ_n(G)}{n}.$$

μ有界图的Bose-Laskar-Hoffman理论

A Bose-Laskar-Hoffman theory for $μ$-bounded graphs with fixed smallest eigenvalue

本文提出结合Bose-Laskar与Hoffman理论的新方法，研究具有固定最小特征值的μ有界图结构，证明参数α在特定条件下受多项式或常数限制。

A lower bound for the smallest eigenvalue of a graph and an application to the associahedron graph

In this paper, we obtain a lower bound for the smallest eigenvalue of a regular graph containing many copies of a smaller fixed subgraph. This generalizes a result of Aharoni, Alon, and Berger in which the subgraph is a triangle. We apply our results to obtain a lower bound on the smallest eigenvalue of the associahedron graph, and we prove that this bound gives the correct order of magnitude of this eigenvalue. We also survey what is known regarding the second-largest eigenvalue of the associahedron graph.

Fat Hoffman graphs with smallest eigenvalue greater than -3

In this paper, we give a combinatorial characterization of the special graphs of fat Hoffman graphs containing $\mathfrak{K}_{1,2}$ with smallest eigenvalue greater than -3, where $\mathfrak{K}_{1,2}$ is the Hoffman graph having one slim vertex and two fat vertices.

Non-bipartite distance-regular graphs with a small smallest eigenvalue

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $θ_{\min}\leq -αk$, where $0<α<1$ is any fixed number. In this paper, we will study non-bipartite distance-regular graphs with relatively small $θ_{\min}$ compared with $k$. In particular, we will show that if $θ_{\min}$ is relatively close to $-k$, then the odd girth $g$ must be large. Also we will classify the non-bipartite distance-regular graphs with $θ_{\min} \leq \frac{D-1}{D}$ for $D =4,5$.

图的最小特征值的组合上界

Combinatorial upper bounds for the smallest eigenvalue of a graph

本研究围绕图的最小特征值展开，基于诱导二部子图给出其组合上界，推导出依赖平均度、色数与独立数的两个上界，引入两个新图不变量，并分析了上界的尖锐性及与现有上界的对比。

On the smallest eigenvalues of the line graphs of some trees

In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetković and Stevanović.

Some observations on the smallest adjacency eigenvalue of a graph

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower b…

The distance-regular graphs such that all of its second largest local eigenvalues are at most one

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graph with smallest eigenvalue $-1-\frac{b_1}{2}$.

On Distance-Regular Graphs with Smallest Eigenvalue at Least $-m$

A non-complete geometric distance-regular graph is the point graph of a partial geometry in which the set of lines is a set of Delsarte cliques. In this paper, we prove that for fixed integer $m\geq 2$, there are only finitely many non-geometric distance-regular graphs with smallest eigenvalue at least $-m$, diameter at least three and intersection number $c_2 \geq 2$.

具有最小特征值至少为-2的图的Hoffman可着色性

Hoffman colorability of graphs with smallest eigenvalue at least -2

本文研究了具有最小特征值至少为-2的图的Hoffman可着色性，扩展了线图的特征化结果，并分类了例外图，同时确定了最大可表示图。

Fat Hoffman graphs with smallest eigenvalue at least $-1-τ$

In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-τ, where τis the golden ratio, can be described by a finite set of fat (-1-τ)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenvalue at least -1-τis an H-line graph, where H is the set of isomorphism classes of maximal fat (-1-τ)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1-τ)-irreducible Hoffman graphs, up to isomorphism.

关于具有4个不同特征值的共边正则图

On co-edge-regular graphs with 4 distinct eigenvalues

该研究通过构造两类具有四个不同特征值且最小特征值固定的无限族连通共边正则图，反驳了 Tan 等人的猜想，并证明 Latin Square 图的团扩张并非谱确定。

最小特征值至少为\(-3\)的符号图及其格

Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices

研究最小特征值至少为\(-3 - \varepsilon\)的连通符号图，证明最小度足够大时最小特征值至少为\(-3\)，相关格是标准格\(\mathbb{Z}^n\)与\(E_8\)根格直和的子格，还讨论了源于无根不可约幺模格的最小特征值为\(-3\)的符号图。

Upper bound for the Laplacian eigenvalues of a graph

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $λ_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest Laplacian eigenvalue of $G$ respectively. Then $ λ_{m+1}(G)\leq d_{m}(G)+m-1 $ for $\bar{G} \neq K_{m}+(n-m)K_1 $. We also introduce upper and lower bound for the Laplacian eigenvalues of weighted graphs, and compare it with the special case of unweighted graphs.

Eigenvalue monotonicity of $q$-Laplacians of trees along a poset

Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.

Eigenvalue bounds for some classes of matrices associated with graphs

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular graphs. Then, we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest eigenvalue and the largest eigenvalue of the Laplacian matrices of graphs. Sharpness of these bounds are verified by examples.

关于具有高奇数圈长的图的最大的和最小的特征值之和

On the sum of the largest and smallest eigenvalues of graphs with high odd girth

该研究通过逼近理论证明了具有高奇数圈长的图的特征值之和上界，给出了γ_k的紧上界至多项对数因子。

无奇数环图的最大与最小特征值之和

On the sum of the largest and smallest eigenvalues of odd-cycle free graphs

本文研究无奇数环图中最大与最小邻接特征值之和，将 Csikvári 对奇围长 5 的界推广到一般奇围长 $k$，并给出奇围长 7 时更紧的数值上界。

A generalization of a theorem of Hoffman

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least $-2$. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every $λ\leq -2$, if a graph with smallest eigenvalue at least $λ$ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph $H(3,q)$, the Johnson graph $J(v, 3)$ and the $2$-clique extension of grids, respectively.

论3-可着色图的最小特征值

On the smallest eigenvalues of $3$-colorable graphs

本文证明3-可着色图的最小特征值在$(-\infty,-\lambda^*)$内稠密，并据此说明球面二距离集研究中无法借助带符号图色数进一步改进禁用子图方法。

Bipartite and neighborhood graphs and the spectrum of the normalized graph Laplacian

We study the spectrum of the normalized Laplace operator of a connected graph $Γ$. As is well known, the smallest nontrivial eigenvalue measures how difficult it is to decompose $Γ$ into two large pieces, whereas the largest eigenvalue controls how close $Γ$ is to being bipartite. The smallest eigenvalue can be controlled by the Cheeger constant, and we establish a dual construction that controls the largest eigenvalue. Moreover, we find that the neighborhood graphs $Γ[l]$ of order $l\geq2$ encode important spectral information about $Γ$ itself which we systematically explore. In particular, t…

Strongly Regular Graphs as Laplacian Extremal Graphs

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also pr…

Making Markov chains less lazy

The mixing time of an ergodic, reversible Markov chain can be bounded in terms of the eigenvalues of the chain: specifically, the second-largest eigenvalue and the smallest eigenvalue. It has become standard to focus only on the second-largest eigenvalue, by making the Markov chain "lazy". (A lazy chain does nothing at each step with probability at least 1/2, and has only nonnegative eigenvalues.) An alternative approach to bounding the smallest eigenvalue was given by Diaconis and Stroock and Diaconis and Saloff-Coste. We give examples to show that using this approach it can be quite easy to …

Neumaier graphs with few eigenvalues

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this paper we give some sufficient and necessary conditions for a Neumaier graph to be strongly regular. Further we show that there does not exist Neumaier graphs with exactly four distinct eigenvalues. We also determine the Neumaier graphs with smallest eigenvalue -2.

超越Cameron、Goethals、Seidel和Shult分类定理

Beyond the classification theorem of Cameron, Goethals, Seidel, and Shult

本文扩展了Cameron等人关于图最小特征值的分类定理，首次对无限多个连通图的最小特征值区间进行完整分类。

The smallest $I_5$-free and triangle-free binary matroids

We determine the smallest simple triangle-free binary matroids that have no five-element independent flat. This solves a special case of a conjecture of Nelson and Norin.

A conjecture of eigenvalues of threshold graphs

Let $A_n$ be the anti-regular graph of order $n.$ It was conjectured that among all threshold graphs on $n$ vertices, $A_n$ has the smallest positive eigenvalue and the largest eigenvalue less than $-1.$ Recently, in \cite{Cesar2} was given partial results for this conjecture and identified the critical cases where a more refined method is needed. In this paper, we deal with these cases and confirm that conjecture holds.

关于具有极值最小特征值的完全图的顶点传递距离正则覆盖

On vertex-transitive distance-regular covers of complete graphs with an extremal smallest eigenvalue

本文研究完全图的阿贝尔距离正则覆盖，利用秩至多3的置换群分类，刻画了具有极值最小特征值且自同构群顶点传递、弧集上至多两个轨道的覆盖族。

Graphs with three distinct distance eigenvalues

In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue $-1$, and determine all $\{C_3,C_4\}$-free connected graphs with three distinct distance eigenvalues of which the smallest one is equal to $-3$, which partially answers a problem posed by Koolen, Hayat and Iqbal [Linear Algebra Appl. 505 (2016) 97--108]. Furthermore, we characterize all trees with three distinct distance eigenvalues.