Qianqian Yang, Brhane Gebremichel, Masood Ur Rehman, Jae Young Yang, Jack H. Koolen
Koolen et al. showed that if a graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will focus on the sesqui-regular graphs with smallest eigenvalue at least $-3$ and study their integrability.
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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.
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。
On the limit points of the smallest eigenvalues of regular graphs
本文给出了半开区间$[-1-\sqrt2, -2)$中具有最小特征值的（非同构）连通$k$ -正则图的无穷多个例子，以及半开区间$[α_1, -1-\sqrt2)$中具有最小特征值的（非同构）连通$k$ -正则图的无穷多个例子，其中$α_1$是多项式$x^3+2x^2-2x-2$的最小根$(\approx -2.4812)$.利用这些结果，我们确定了小于-2的正则图的最小特征值的最大极限点和次大极限点。此外，我们还确定了最小特征值小于-2的连通3-正则图的最小特征值的上确界，并给出了以这个上确界为最小特征值的唯一图.
The smallest eigenvalue of K_p-free graphs
本文给出了无p-团图的邻接矩阵的最小特征值的一个上界。
In this paper, we show that a connected graph with smallest eigenvalue at least -3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least -2.
In this paper, we study the non-bipartite distance-regular graphs with valency k and having a smallest eigenvalue at most -k/2.
We give a structural classification of edge-signed graphs with smallest eigenvalue greater than -2. We prove a conjecture of Hoffman about the smallest eigenvalue of the line graph of a tree that was stated in the 1970s. Furthermore, we prove a more general result extending Hoffman's original statement to all edge-signed graphs with smallest eigenvalue greater than -2. Our results give a classification of the special graphs of fat Hoffman graphs with smallest eigenvalue greater than -3.
A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Non-bipartite distance-regular graphs with a small smallest eigenvalue, Electronic J. Combin. 26(2) (2019), P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.
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…
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
对于具有第二大特征值的距离正则图（分别为最小特征值）\mu 1（分别\muD）我们证明了（\mu 1+1）（\muD +1）<= -b1成立，其中等式仅在直径等于2时成立。利用这个不等式，我们研究了具有固定第二大特征值的距离正则图。
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。
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
本文证明了图的正规拉普拉斯算子λ 1的调和本征函数在图的铰接点上的单调性定理。本文给出了图的正规拉普拉斯矩阵的Perron分量的定义，并给出了它的次小特征值的刻画。
On the Smallest Eigenvalue of Grounded Laplacian Matrices
我们提供了上界和下界上的最小特征值的接地拉普拉斯矩阵（这是通过删除特定的行和列的拉普拉斯矩阵的给定图获得的矩阵）。上界和下界之间的间隙取决于与接地拉普拉斯算子的最小特征值相对应的特征向量的最小和最大分量的比率。我们提供了一个图论上的这个比率的界限，并随后获得了严格的特征值的最小特征值的某些类别的图。具体来说，对于Erdos Renyi随机图，我们表明，当一个（足够小）集$S$的行和列从拉普拉斯算子，并添加一个边缘的概率$p$是足够大的，接地拉普拉斯算子的最小特征值渐近几乎肯定接近$|S|p$。我们还表明，随机$d$ -正则图与一个单一的行和列删除，最小的特征值是$Θ(\frac{d}{n})$。我们的边界有应用程序的收敛速度在连续时间和离散时间的一致性动力学与顽固或领导节点的研究。
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。
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。
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
这是同一标题的文章的延续。在本文中，族H是相同的，在以前的文件“与最小特征值至少$-1-\sqrt{2}$，第一部分”的图。主要结果是：非H线图的极小图与计算机所找到的38个图之一同构。
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.
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。
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.
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.
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.
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
利用胖Hoffman图的特殊图，研究了最小特征值至少为-3的胖Hoffman图.证明了不可分解胖Hoffman图H的特殊图S（H）可用标准格或不可约根格表示。此外，我们还证明了如果特殊图S（H）具有整表示，即它所张成的格不是例外根格，则特殊图S（H）同构于Dynkin图A_n，D_n或扩展Dynkin图A_n或D_n之一.
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 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。
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…
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 $$\frac{μ_1(G)+μ_n(G)}{n}.$$
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。
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 $D =4,5$.
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。
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.
The distance-regular graphs such that all of its second largest local eigenvalues are at most one
本文对距离正则图进行了分类，使得它的第二大局部特征值至多为1。并讨论了距离正则图的最小特征值的性质。推广了第一作者对具有最小特征值的距离正则图进行分类的结果$-1-\frac{b_1}{2}$。
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。
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.
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。
Fat Hoffman graphs with smallest eigenvalue greater than -3
本文给出了含有最小特征值大于-3的$\mathfrak{K}_{1,2}$的胖Hoffman图的特殊图的组合刻划，其中$\mathfrak{K}_{1,2}$是具有一个瘦点和两个胖点的Hoffman图.
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.
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…
Upper bound for the Laplacian eigenvalues of a graph
本文给出了无权图的Laplacian特征值的一个新的上界。设$G$是以$n$为顶点的单图.令$d_{m}(G)$和$λ_{m+1}(G)$分别为$G$的第$m$个最小度和$G$的第$m+1$个最小拉普拉斯特征值。然后为$\bar{G} \neq K_{m}+(n-m)K_1 $选择$ λ_{m+1}(G)\leq d_{m}(G)+m-1 $。我们还引入了赋权图的Laplacian特征值的上、下界，并将其与特殊情况的非赋权图进行了比较。
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.
On Distance-Regular Graphs with Smallest Eigenvalue at Least $-m$
非完全几何距离正则图是部分几何中的点图，其中直线集是Delsarte团集。本文证明了对于固定整数$m\geq 2$，最小特征值至少为$-m$，直径至少为3，相交数$c_2 \geq 2$的非几何距离正则图的个数仅为1/2.
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。
Fat Hoffman graphs with smallest eigenvalue at least $-1-τ$
本文证明了所有最小特征值至少为-1-τ（τ为黄金比例）的胖Hoffman图都可以用胖（-1-τ）-不可约Hoffman图的有限集来描述.用Woo和Neumaier的术语，我们认为每个最小特征值至少为-1-τ的胖Hoffman图都是H线图，其中H是极大胖（-1-τ）-不可约Hoffman图的同构类的集合。结果表明，直到同构为止，存在37个胖（-1-τ）-不可约Hoffman图.
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。
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.
Bipartite and neighborhood graphs and the spectrum of the normalized graph Laplacian
研究了连通图$Γ$的正规化Laplace算子的谱.众所周知，最小的非平凡特征值测量将$Γ$分解成两大块的难度，而最大的特征值控制$Γ$与二分的接近程度。最小的特征值可以控制的Cheeger常数，我们建立了一个对偶结构，控制最大的特征值。此外，我们发现的顺序$l\geq2$的邻域图$Γ[l]$编码重要的频谱信息$Γ$本身，我们系统地探讨。特别是，邻域图方法导致新的估计，可以改善Cheeger不等式的最小的非平凡特征值，以及明确估计的最大特征值从上面和下面。作为这种谱估计的应用，我们给出了耦合映象格子的同步性的一个判据，以及图上随机游动的收敛速度的一个估计。
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。
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…
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.
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。
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.
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。
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.
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。
Making Markov chains less lazy
一个遍历的可逆马尔可夫链的混合时间可以用链的特征值来定界，具体地说，就是第二大特征值和最小特征值。通过使马尔可夫链"懒惰"，只关注第二大特征值已经成为标准。（惰性链在每一步都不做任何事情，概率至少为1/2，并且只有非负特征值。Diaconis和Stroock以及Diaconis和Saloff-Coste给出了另一种确定最小特征值的方法。我们给出的例子表明，使用这种方法，它可以很容易地获得一个组合马尔可夫链的最小特征值的约束，这是几个数量级低于最知名的第二大特征值的约束。
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.
On the sign patterns of the smallest signless Laplacian eigenvector
设$H$是连通二部图，其无符号拉普拉斯矩阵为$Q(H)$ .设$H$的二分划是$(S,T)$，$x$是$Q(H)$的最小特征值的特征向量.众所周知，$x$在$S$上为正且恒定，在$T$上为负且恒定。本文研究了由$S$或$T$引入的子图中添加边时$x$的符号模式的恢复性，并描述了$x$的符号模式持续存在的一些情况。
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