Michael Krivelevich, Benny Sudakov
We prove that for all values of the edge probability p(n) the largest eigenvalue of a random graph G(n,p) satisfies almost surely: λ_1(G)=(1+o(1))max{\sqrtΔ,np}, where Δis a maximal degree of G, and the o(1) term tends to zero as max{\sqrtΔ,np} tends to infinity.
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Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely λ_1(G)= (1+o(1)) max(Δ^{1/2}(G),np), where Δ(G) is the maximum degree of G and o(1) term tends to zero as max (Δ^{1/2}(G), np) tends to infinity.
We classify the connected graphs with precisely three distinct eigenvalues and second largest eigenvalue at most 1.
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.
We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq λ_N\leq 2$, to the case of chemical hypergraphs.
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.
We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.
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。
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 $-1-\frac{b_1}{2}$.
We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem.
We consider a sparse random subraph of the $n$-cube where each edge appears independently with small probability $p(n) =O(n^{-1+o(1)})$. In the most interesting regime when $p(n)$ is not exponentially small we prove that the largest eigenvalue of the graph is asymtotically equal to the square root of the maximum degree.
In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erdős-Rényi random graph $G(n,p)$, answering a question raised by Dekel-Lee-Linial.
The upper tail problem for the largest eigenvalue of the Erdős--Rényi random graph $\mathcal{G}_{n,p}$ is to estimate the probability that the largest eigenvalue of the adjacency matrix of $\mathcal{G}_{n,p}$ exceeds its typical value by a factor of $1+δ$. In this note we show that for $δ>0$ fixed, and $p \rightarrow 0$ such that $n^{\frac{1}{2}} p \rightarrow \infty$, the upper tail probability for the largest eigenvalue of $\mathcal{G}_{n,p}$ is $$\exp\left[-(1+o(1)) \min\left\{\tfrac{(1+δ)^2}{2}, δ(1+δ) \right\} n^{2}p^{2}\log (1/p)\right].$$ In the same regime of $p$, we show that the seco…
The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.
We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.
We define a new Cheeger-like constant for graphs and we use it for proving Cheeger-like inequalities that bound the largest eigenvalue of the normalized Laplace operator.
Let $v(k, λ)$ be the maximum number of vertices of a connected $k$-regular graph with second largest eigenvalue at most $λ$. The Alon-Boppana Theorem implies that $v(k, λ)$ is finite when $k > \frac{λ^2 + 4}{4}$. In this paper, we show that for fixed $λ\geq1$, there exists a constant $C(λ)$ such that $2k+2 \leq v(k, λ) \leq 2k + C(λ)$ when $k > \frac{λ^2 + 4}{4}$.
Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs
本文研究有向图的最大特征值和拉普拉斯特征值之和的界，证明了非二部图包含平衡三角形的条件，并验证了一个关于拉普拉斯特征值与顶点度数关系的猜想。
For every real $0\leq α\leq 1$, Nikiforov defined the $A_α$-matrix of a graph $G$ as $A_α(G)=αD(G)+(1-α)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. The eigenvalues of $A_α(G)$ are called the $A_α$-eigenvalues of $G$. Let $S_k(A_α(G))$ be the sum of $k$ largest $A_α$-eigenvalues of $G$. In this paper, we present several upper and lower bounds on $S_k(A_α(G))$ and characterize the extremal graphs for certain cases, which can be regard as a common generalization of the sum of $k$ largest eigenvalues of adjacency matrix and s…
We investigate the structure of trees that have greatest maximum eigenvalue among all trees with a given degree sequence. We show that in such an extremal tree the degree sequence is non-increasing with respect to an ordering of the vertices that is obtained by breadth-first search. This structure is uniquely determined up to isomorphism. We also show that the maximum eigenvalue in such classes of trees is strictly monotone with respect to majorization.
Tricyclic graphs for which the second largest distance eigenvalue less than $-\frac{1}{2}$
本文基于距离矩阵第二大特征值小于$-\frac{1}{2}$的图论性质，完整刻画了满足该条件的所有三环图。
Connected triangle-free planar graphs whose second largest eigenvalue is at most 1
本文利用谱理论和局部结构刻画，完全确定了所有不含 $C_3$ 且第二大特征值不超过1的连通 $\{K_5,K_{3,3}\}$-minor free 图，部分解决了Li和Sun提出的关于连通平面图特征值的开放问题。
Let $G$ be a connected graph and let $T$ be a spanning tree of $G$. A partial orientation $σ$ of $G$ respect to $T$ is an orientation of the edges of $G$ except those edges of $T$, the resulting graph associated with which is denoted by $G_T^σ$. In this paper we prove that there exists a partial orientation $σ$ of $G$ respect to $T$ such that the largest eigenvalue of the Hermitian adjacency matrix of $G_T^σ$ is at most the largest absolute value of the roots of the matching polynomial of $G$.
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.
On the second largest adjacency eigenvalue of trees with given diameter
本文在给定顶点数和直径的树类中刻画了第二大邻接特征值的极值树，并重新讨论了树的谱中心及λ₂最大化证明，丰富了图族λ₂极值化研究。
In this paper, we show that the largest signless Laplacian H-eigenvalue of a connected $k$-uniform hypergraph $G$, where $k \ge 3$, reaches its upper bound $2Δ(G)$, where $Δ(G)$ is the largest degree of $G$, if and only if $G$ is regular. Thus the largest Laplacian H-eigenvalue of $G$, reaches the same upper bound, if and only if $G$ is regular and odd-bipartite. We show that an $s$-cycle $G$, as a $k$-uniform hypergraph, where $1 \le s \le k-1$, is regular if and only if there is a positive integer $q$ such that $k=q(k-s)$. We show that an even-uniform $s$-path and an even-uniform non-regular…
We study the eigenspace of the Laplacian matrix of a simple graph corresponding to the largest eigenvalue, subsequently arriving at the theory of modular decomposition of T. Gallai.
A graph for which the second largest distance eigenvalue is less than $\frac{-3+\sqrt{5}}{2}$ is chordal
本文定义了图的距离矩阵及其第二大距离特征值，证明了该特征值小于$\frac{-3+\sqrt{5}}{2}$的连通图必为弦图，并刻画了特征值小于$-\frac{1}{2}$的双环图与分裂图。
We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when $d$ is sufficiently large. Our proofs rely on the method of compressions.
Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $λ_{1}(D(G))\geqλ_{2}(D(G))\geq\cdots\geqλ_{n}(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral characterization on the second largest distance eigenvalue $λ_{2}(D(G))$ of graphs, and prove that the graphs with $λ_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by their $D$-spectra.
In this article, we study Pareto eigenvalues of distance matrix of connected graphs and show that the non zero entries of every distance Pareto eigenvector of a tree forms a strictly convex function on the forest generated by the vertices corresponding to the non zero entries of the vector. Besides we find the minimum number of possible distance Pareto eigenvalue of a connected graph and establish lower bounds for $n$ largest distance Pareto eigenvalues of a connected graph of order $n.$ Finally, we discuss some bounds for the second largest distance Pareto eigenvalue and find graphs with opti…
Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value of the largest non-trivial eigenvalue is at most 2 sqrt{d-1} then the graph is said to be Ramanujan. These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, coding theory and cryptography. As many of these applications depend on the size of the largest non-trivial po…
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。
We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $λ_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$), we have $$ M(G) \le λ_{\max}(G) \le \big(\frac14 \log|V| + 1 \big) \M(G). $$ Here the lower bound is attained if $G$ is regular or bi-regular, whereas the logarithmic factor in the upper bound, conjecturally, can be improved --- although we present an example showing that it cannot be replaced with a factor growing slower than $(\log |V|/\log\log|V|)^{1/8}$. Furthe…
Characterizing graphs with the second largest distance eigenvalue less than -1/2
本文通过谱和结构方法研究了第二大的距离特征值小于-1/2的连通图的刻画问题。
Let $A_n$ denote the alternating group of degree $n$ with $n\geq 3$. The alternating group graph $AG_n$, extended alternating group graph $EAG_n$ and complete alternating group graph $CAG_n$ are the Cayley graphs $\mathrm{Cay}(A_n,T_1)$, $\mathrm{Cay}(A_n,T_2)$ and $\mathrm{Cay}(A_n,T_3)$, respectively, where $T_1=\{(1,2,i),(1,i,2)\mid 3\leq i\leq n\}$, $T_2=\{(1,i,j),(1,j,i)\mid 2\leq i<j\leq n\}$ and $T_3=\{(i,j,k),(i,k,j)\mid 1\leq i<j<k\leq n\}$. In this paper, we determine the second largest eigenvalues of $AG_n$, $EAG_n$ and $CAG_n$.
The eccentricity matrix $\varepsilon(G)$ of a graph $G$ is constructed from the distance matrix of $G$ by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its eccentricity matrix $\varepsilon(G)$. Wang et al \cite{e} proposed the problem of determining the maximum $\varepsilon$-spectral radius of trees with given order. In this paper, we consider t…
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.
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 $$\frac{μ_1(G)+μ_n(G)}{n}.$$
At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian
本文研究归一化拉普拉斯矩阵最大特征值与着色数之间的色数界，刻画了达到下界 χ/(χ-1) 的图及其重数，并给出相关上界。
In July 2012, at the Conference on Applications of Graph Spectra in Computer Science, Barcelona, D. Stevanovic posed the following open problem: which graphs have the zero as the largest eigenvalue of their modularity matrix? The conjecture was that only the complete and complete multipartite graphs. They indeed have this property, but are they the only ones? In this paper, we will give an affirmative answer to this question and prove a bit more: both the modularity and the normalized modularity matrix of a graph is negative semidefinite if and only if the graph is complete or complete multipa…
We show that for any $d=d(n)$ with $d_0(ε) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm ε)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory.
Sum of the $k$ Largest Eigenvalues of Symmetric Matrices: Theory and Applications
本文建立了对称矩阵的 $k$ 个最大特征值之和的新上界，改进了Mohar的相关界，并证明了对几乎所有图，Brouwer猜想在小$k$情况下成立。
A lower bound of toughness of regular graphs: in terms of second largest eigenvalue
研究正则图的 toughness 与第二大的特征值的关系，证明 toughness 下界与特征值有关。
For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=λ_1(D(G))+\cdots+λ_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and lower bound of $S_k(D(G))$. We determine the sharp lower bounds of $S_k(D(G))$ when $G$ is connected graph and is a tree, respectively, and characterize both the extremal graphs. Moreover, we conjecture that the upper bound is attained when $G$ is a path of order $n$ and prove some partial result supporting the conjecture. To prove our result, we obtain a sharp upper boun…
Large deviations of the largest eigenvalue of supercritical sparse Wigner matrices
针对平均度至少为维度对数阶的超临界稀疏Wigner矩阵，研究其最大特征值的大偏差上尾事件，揭示由高度顶点或大权重重团引发的机制，发现速率函数在典型值处不连续，补充了常平均度情形的已有结果。
The Second Largest Eigenvalue of Stiffness Matrices of Normalized Complete Frameworks
研究顶点在单位球面上且质心在原点的完全框架\(L(K_n,p)\)的第二大特征值，证明当\(d\geq2\)且\(p\)像含至少三点时该值为\(n/2\)，还构造例子表明猜想中重数不正确，揭示特征值与重数对底层点配置的不同特性。
On the sum of the first two largest signless Laplacian eigenvalues of a graph
本文证明 Oliveira 等人关于 $n$ 顶点图中加边星形图唯一最小化 $f(G)$ 的猜想，并由此确认 Ashraf 等人提出的 $S_2(G)$ 上界猜想。
Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues
本文研究了前两个最大无符号拉普拉斯特征值之和的极图问题，证明了在具有$e(G)$条边的图中，$K^+_{1,e(G)-1}$是使$f(G)$取得最小值的唯一图。
Improved lower bounds on the extrema of eigenvalues of graphs
本文通过二十面体图的闭膨胀，改进了图的第四大特征值最大值的下界，回答了Nikiforov的问题。
The second largest eigenvalue of normal Cayley graphs on symmetric groups generated by cycles
本文研究由指定长度循环生成的正规凯莱图的第二大特征值，证明其至多由四个不可约表示达到，并据此刻画 Aldous 性质及证明相关猜想。
The largest Laplacian eigenvalue and the balancedness of simplicial complexes
本文证明单纯复形上拉普拉斯最大特征值不超过上无符号拉普拉斯最大特征值，并给出等号成立与关联符号图平衡性的等价条件，同时改进特征值上界并刻画多种运算下的平衡性。
Let $\mathcal{A(}G\mathcal{)},\mathcal{L(}G\mathcal{)}$ and $\mathcal{Q(}% G\mathcal{)}$ be the adjacency tensor, Laplacian tensor and signless Laplacian tensor of uniform hypergraph $G$, respectively. Denote by $λ(\mathcal{T})$ the largest H-eigenvalue of tensor $\mathcal{T}$. Let $H$ be a uniform hypergraph, and $H^{\prime}$ be obtained from $H$ by inserting a new vertex with degree one in each edge. We prove that $λ(\mathcal{Q(}% H^{\prime}\mathcal{)})\leqλ(\mathcal{Q(}H\mathcal{)}).$ Denote by $G^{k}$ the $k$th power hypergraph of an ordinary graph $G$ with maximum degree $Δ\geq2$. We will…
The largest Laplacian eigenvalue of induced-$K_{1,r}$-free graphs
研究不含诱导子图\(K_{1,k}\)的图的最大拉普拉斯特征值问题，通过结合拉普拉斯瑞利向量符号分解与独立集加权局部卡罗 - 魏不等式，证明\(\mu(G)\leq(2-\frac{2}{k})(d + 1)\)，给出该问题更强精确界。
Let m and r be two integers. Let G be a connected r-regular graph of order n and k an integer depending on m and r. For even kn, we find a best upper bound (in terms of r and m) on the third largest eigenvalue that is sufficient to guarantee that G has a k-factor. When nk is odd, we give a best upper bound (in terms of r and m) on the second largest eigenvalue that is sufficient to guarantee that G is k-critical.
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 …
The largest eigenvalue of $\mathcal{C}_4^{-}$-free signed graphs
本文研究了无 $\mathcal{C}_4^{-}$ 不平衡符号图的最大指数，完全刻画了极值图，并将其视为 Nikiforov 及 Zhai 与 Wang 相关结果的符号图版本。
Let G be a simple graph on $n$ vertices and $e(G)$ edges. Consider $Q(G) = D + A$ as the signless Laplacian of $G$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of the vertices degree of $G$. Let $q_1(G)$ and $q_2(G)$ be the first and the second largest eigenvalues of $Q(G),$ respectively, and denote by $S_{n}^{+}$ the star graph plus one edge. In this paper, we prove that inequality $q_1(G)+ q_2(G) <= e(G)+3$ is tighter for the graph $S_{n}^{+}$ among all firefly graphs and also tighter to $S_{n}^{+}$ than to the graphs $K_{k} \vee \overline{K_{n-k}}$ recently presented by…
In this paper, we introduce the class of cored hypergraphs and power hypergraphs, and investigate the properties of their Laplacian H-eigenvalues. From an ordinary graph, one may generate a $k$-uniform hypergraph, called the $k$th power hypergraph of that graph. Power hypergraphs are cored hypergraphs, but not vice versa. Hyperstars, hypercycles, hyperpaths are special cases of power hypergraphs, while sunflowers are a subclass of cored hypergraphs, but not power graphs in general. We show that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signl…
The spectral theory of higher-order symmetric tensors is an important tool to reveal some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop a first-order optimization algorithm for computing H- and Z-eigenvalues of these large scale sparse tensors (CEST). With the aid of the Kurdyka-Łojasiewicz property, we prove that the sequence of iterates generated…
The second largest eigenvalue of some nonnormal Cayley graphs on symmetric groups
本文证明对称群上由移动前r个点的k-循环生成的非正规Cayley图具有Aldous性质，即严格第二大特征值由标准表示取得，仅排除两类例外情况，并确定了相关图的特征值对应的不可约表示。
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