Graphs with three eigenvalues and second largest eigenvalue at most 1

Graphs with second largest eigenvalue less than $1/2$

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.

Minimum supports of eigenfunctions with the second largest eigenvalue of the Star graph

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.

Largest regular multigraphs with three distinct eigenvalues

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.

On the order of regular graphs with fixed second largest eigenvalue

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 $k > \frac{λ^2 + 4}{4}$.

有向图最大特征值和拉普拉斯特征值之和的界

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。

On the sum of the largest $A_α$-eigenvalues of graphs

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…

Largest Laplacian Eigenvalue and Degree Sequences of Trees

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.

第二大距离特征值小于$-\frac{1}{2}$的三环图

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。

第二大特征值至多为1的连通无三角形平面图

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。

On the largest eigenvalue of a mixed graph with partial orientation

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 $G$.

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.

关于给定直径树类的第二大邻接特征值

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。

Regular Uniform Hypergraphs, $s$-Cycles, $s$-Paths and Their largest Laplacian H-Eigenvalues

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…

一个第二大距离特征值小于$\frac{-3+\sqrt{5}}{2}$的图是弦图

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。

Eigenvalues of subgraphs of the cube

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.

On the second largest distance eigenvalue of a graph

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.

On Pareto eigenvalue of distance matrix of graphs

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…

The Distribution of the Largest Non-trivial Eigenvalues in Families of Random Regular Graphs

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…

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 Largest Eigenvalue and Bi-Average Degree of a Graph

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…

用第二大的距离特征值小于-1/2来刻画图

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。

The second largest eigenvalues of some Cayley graphs on alternating groups

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 $CAG_n$.

On the largest and least eigenvalues of eccentricity matrix of trees

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…

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 $$\frac{μ_1(G)+μ_n(G)}{n}.$$

Cheeger-like inequalities for the largest eigenvalue of the graph Laplace Operator

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.

对称矩阵的 $k$ 个最大特征值之和：理论与应用

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。

正则图 toughness 的下界：以第二大的特征值为参数

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。

THe largest eigenvalue of sparse random graphs

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.

On the sum of $k$-th largest distance eigenvalues of graphs

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…

归一化完全框架刚度矩阵的第二大特征值

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。

关于图的前两大无符号拉普拉斯特征值之和

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。

前两个最大无符号拉普拉斯特征值之和的极图

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。

图的特征值极值的下界改进

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。

由循环生成的对称群上正规凯莱图的第二大特征值

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。

单纯复形的最大拉普拉斯特征值与平衡性

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。

The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs

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…

Largest Eigenvalue of the Laplacian Matrix

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.

无诱导子图\(K_{1,r}\)的图的最大拉普拉斯特征值

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。

Regular factors of regular graphs from eigenvalues

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.

On the Largest Eigenvalue of a Random Subgraph of the Hypercube

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.

无 $\mathcal{C}_4^{-}$ 符号图的最大特征值

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。

对称群上某些非正规Cayley图的第二大特征值

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。

单纯复形的最大归一化拉普拉斯特征值与关联平衡性

The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes

本文利用带符号图的平衡性，刻画了单纯复形上归一化拉普拉斯算子最大特征值取到 $i+2$ 的充要条件及其重数，并构造了无穷多类满足该条件的单纯复形。

On the limit points of the smallest eigenvalues of regular graphs

In this paper, we give infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[-1-\sqrt2, -2)$ and also infinitely many examples of (non-isomorphic) connected $k$-regular graphs with smallest eigenvalue in half open interval $[α_1, -1-\sqrt2)$ where $α_1$ is the smallest root$(\approx -2.4812)$ of the polynomial $x^3+2x^2-2x-2$. From these results, we determine the largest and second largest limit points of smallest eigenvalues of regular graphs less than -2. Moreover we determine the supremum of the smallest eigenvalue among a…

关于图上旋度-旋度算子的两个最大特征值之和

On the sum of the two largest eigenvalues of the curl-curl operator on graphs

本文证明了图上的旋度-旋度算子两个最大特征值之和不超过第二阶度数序列共轭的前两项之和，验证了Duval-Reiner猜想的前两个优超不等式。

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.

On the largest eigenvalue of a sparse random subgraph of the hypercube

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.

幂超图的特征值中第二大模

On the second-largest modulus among the eigenvalues of a power hypergraph

本文确定了幂超图特征值中第二大模，并证明其对应射影特征簇维数为零，且代数重数与几何重数相等，用最弱边数表示。

Chromatic number and spectral radius

Let G be a graph, H be its chromatic number, L be the largest eigenvalue of its Laplacian, and M be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that L>=(H/(H-1))M

谱的尽头：归一化拉普拉斯矩阵最大特征值的色数界

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。

When the largest eigenvalue of the modularity and normalized modularity matrix is zero

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…

Eigenvalues of Non-Regular Linear-Quasirandom Hypergraphs

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.

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 …

Extremal graphs for the sum of the two largest signless Laplacian eigenvalues

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…