Vladimir Nikiforov
Given a graph G of order n and size m, let s(G)= sum|d(u)-2m/n|, where the sum is taken over all vertices u of G. We investigate upper and lower bounds on eigenvalues of G in terms of s(G).
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Let $G$ be a graph. In a famous paper Collatz and Sinogowitz had proposed to measure its deviation from regularity by the difference of the (adjacency) spectral radius and the average degree: $ε(G)=ρ(G)-\frac{2m}{n}$. We obtain here a new upper bound on $ε(G)$ which seems to consistently outperform the best known upper bound to date, due to Nikiforov. The method of proof may also be of independent interest, as we use notions from numerical analysis to re-cast the estimation of $ε(G)$ as a special case of the estimation of the difference between Rayleigh quotients of proximal vectors.
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. We consider the corresponding eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the ergodic sums is bounded for every point in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3, Theorem 1.9).
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.
Bounds on median eigenvalues of graphs of bounded degree
研究证明了最大度数为d的图的中位特征值上界为√(d-1)，并针对三种情况给出了下界，解决了Mohar关于图中位特征值的问题。
Let G be an n-vertex graph with m edges. The degree deviation measure of G is defined as s(G)=sum v in V(G)|degG(v)-(2m/n)|, where n and m are the number of vertices and edges of G, respectively. The aim of this paper is to prove the Conjecture 4.2 of [J A de Oliveira, C S Oliveira, C Justel and N M Maia de Abreu, Measures of irregularity of graphs, Pesq. Oper. 33 (3) (2013) 383-398]. The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed.
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。
Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both …
Eigenvalues and spanning trees with constrained degree
本文基于特征值研究了具有约束度数和叶子度数的生成树，为正则图、连通图和一般图分别给出了保证特定生成树存在的谱充分条件，并推广了已有结果。
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…
An Optimization Approach to Degree Deviation and Spectral Radius
本文通过平滑技术将图的度偏差和最大特征值与低维非线性优化问题相联系，在给定图的度数条件下，推导出了度偏差的上界及最大特征值的下界。
We prove a large deviation result for a random symmetric n x n matrix with independent identically distributed entries to have a few eigenvalues of size n. If the spectrum S survives when the matrix is rescaled by a factor of n, it can only be the eigenvalues of a Hilbert-Schmidt kernel k(x,y) on [0,1] x [0,1]. The rate function for k is $I(k)=1/2\int h(k(x,y) dxdy$ where h is the Cramer rate function for the common distribution of the entries that is assumed to have a tail decaying faster than any Gaussian. The large deviation for S is then obtained by contraction.
We improve recent results relating graph eigenvalues to other graph parameters like girth, domination number, and minimum degree.
Degree Deviation and Spectral Radius
本文研究了有限简单无向图的度偏差与最大特征值关系，推进了 Nikiforov 的猜想，证明 $λ-\frac{2m}{n}\leq \sqrt{\frac{2s}{3}}$，并推广了 Stanley 等人的相关结论。
Motivated by the problem about HOMO-LUMO separation that arises in mathematical chemistry, Fowler and Pisanski introduced the notion of the HL-index which measures how large in absolute value may be the median eigenvalues of a graph. In this note we provide rather tight lower and upper bounds on the maximum value of the HL-index among all graphs with given average degree. In particular, we determine the exact value of this parameter when restricted to chemically relevant graphs, i.e.\ graphs of maximum degree $3$, and thus answer a question of Fowler and Pisanski. The proof provides additional…
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.
We show that the number of Laplacian eigenvalues greater than the average degree of a tree having $n$ vertices is at most $\lfloor\frac{n}{2} \rfloor$.
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。
Eigenvalues of Wigner matrices has been a major topic of investigation. A particularly important subclass of such random matrices is formed by the adjacency matrix of an Erdős-Rényi graph $\mathcal{G}_{n,p}$ equipped with i.i.d. edge-weights. An observable of particular interest is the largest eigenvalue. In this paper, we study the large deviations behavior of the largest eigenvalue of such matrices, a topic that has received considerable attention over the years. We focus on the case $p = \frac{d}{n}$, where most known techniques break down. So far, results were known only for $\mathcal{G}_{…
The Large Deviation Principle for $W$-random spectral measures
本文基于$W$-随机图的LDP证明随机对称Hilbert–Schmidt积分算子谱测度的LDP，揭示graphon大偏差对特征值和特征空间的影响。
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In addition, some new unpublished results and questions are concluded. Emphasis is given on classifications of the upper and…
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
A tight upper bound of spectral radius in terms of degree deviation
本文研究图的谱半径上界问题，去掉了Nikiforov猜想中m、n需足够大的前提，证明了用度偏差表示的谱半径与平均度差值的紧上界。
In this paper we consider the problem of estimating the joint upper and lower tail large deviations of the edge eigenvalues of an Erdős-Rényi random graph $\mathcal{G}_{n,p}$, in the regime of $p$ where the edge of the spectrum is no longer governed by global observables, such as the number of edges, but rather by localized statistics, such as high degree vertices. Going beyond the recent developments in mean-field approximations of related problems, this paper provides a comprehensive treatment of the large deviations of the spectral edge in this entire regime, which notably includes the well…
Large deviation behavior of the largest eigenvalue $λ_1$ of Gaussian networks (Erdős-Rényi random graphs $\mathcal{G}_{n,p}$ with i.i.d. Gaussian weights on the edges) has been the topic of considerable interest. Recently in [6,30], a powerful approach was introduced based on tilting measures by suitable spherical integrals, particularly establishing a non-universal large deviation behavior for fixed $p<1$ compared to the standard Gaussian ($p=1$) case. The case when $p\to 0$ was however completely left open with one expecting the dense behavior to hold only until the average degree is logarit…
Preferential attachment schemes, where the selection mechanism is linear and possibly time-dependent, are considered, and an infinite-dimensional large deviation principle for the sample path evolution of the empirical degree distribution is found by Dupuis-Ellis-type methods. Interestingly, the rate function, which can be evaluated, contains a term which accounts for the cost of assigning a fraction of the total degree to an "infinite" degree component, that is, when an atypical "condensation" effect occurs with respect to the degree structure. As a consequence of the large deviation results,…
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…
Distribution of signless Laplacian eigenvalues and degree sequence
本文通过度序列刻画了图的无符号拉普拉斯特征值在特定子区间内的分布，确定了满足某些特征值计数条件的图，并提出了两个开放问题。
We consider large random trees under Gibbs distributions and prove a Large Deviation Principle (LDP) for the distribution of degrees of vertices of the tree. The LDP rate function is given explicitly. An immediate consequence is a Law of Large Numbers for the distribution of vertex degrees in a large random tree. Our motivation for this study comes from the analysis of RNA secondary structures.
Splitting fields and spectral invariants of character degree graphs in solvable groups
本文研究可解群特征标度图的谱算术性质，推导(n-2)-正则图的特征多项式公式，分析超图、Lewis图及直径3图的分裂域与特征值相关性质。
Given a finite group $G$, the character graph, denoted by $Δ(G)$, for its irreducible character degrees is a graph with vertex set $ρ(G)$ which is the set of prime numbers that divide the irreducible character degrees of $G$, and with $\{p,q\}$ being an edge if there exist a non-linear $χ\in {\rm Irr}(G)$ whose degree is divisible by $pq$. In this paper, we discuss the influences of cut vertices and eigenvalues of $Δ(G)$ on the group structure of $G$. Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of charac…
Given a connected graph $G$, the toughness $τ_G$ is defined as the minimum value of the ratio $|S|/ω_{G-S}$, where $S$ ranges over all vertex cut sets of $G$, and $ω_{G-S}$ is the number of connected components in the subgraph $G-S$ obtained by deleting all vertices of $S$ from $G$. In this paper, we provide a lower bound for the toughness $τ_G$ in terms of the maximum degree, minimum degree and normalized Laplacian eigenvalues of $G$. This can be viewed as a slight generalization of Brouwer's toughness conjecture, which was confirmed by Gu (2021). Furthermore, we give a characterization of th…
For every constant $d \geq 3$ and $ε> 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Θ(n)$ vertices that is $ε$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + ε$ (excluding the single trivial eigenvalue of~$d$).
In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $Δ_1 \geq Δ_2 \geq .. \geq Δ_k$ be the degrees of the $k$ highest degree vertices. We prove that at time $t$, for any function $f$ with $f(t) \rightarrow +\infty$ as $t \rightarrow +\infty$, $\frac{t^{1/2}}{f(t)} \leq Δ_1 \leq f(t)t^{1/2}$ and for $i=2,...,k=O(1)$, $\frac{t^{1/2}}{f(t)} \leq Δ_i \leq Δ_{i-1} - \frac{t^{1/2}}{f(t)}$ with high probability (\whp). Then, we show that the…
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