Yonatan Bilu
Hofmman's bound on the chromatic number of a graph states that $χ\geq 1 - \frac {λ_1} {λ_n}$. Here we show that the same bound, or slight modifications of it, hold for several graph parameters related to the chromatic number: the vector coloring number, the $ψ$-covering number and the $λ$-clustering number.
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Let $χ(G)$ denote the chromatic number of a graph and $χ_v(G)$ denote the vector chromatic number. For all graphs $χ_v(G) \le χ(G)$ and for some graphs $χ_v(G) \ll χ(G)$. Galtman proved that Hoffman's well-known lower bound for $χ(G)$ is in fact a lower bound for $χ_v(G)$. We prove that two more spectral lower bounds for $χ(G)$ are also lower bounds for $χ_v(G)$. We then use one of these bounds to derive a new characterization of $χ_v(G)$.
Hoffman colorability of (strongly) regular graphs
该研究将霍夫曼界视为参数，探究其着色（霍夫曼着色）在正则图、（余）边正则图和强正则图中的应用，证明强正则图的霍夫曼着色性蕴含伪几何性，强化相关有限性结果并得到新刻画。
Hoffman colorings of graphs
本文研究非正则图的Hoffman界紧性，提出分解定理与组合定理以刻画Hoffman着色结构，完全分类锥图和线图的Hoffman可着色性，并导出计算连通Hoffman可着色图的算法。
Tales of Hoffman: from a distance
本文扩展了霍夫曼界限到距离-$k$ 的场景，并展示了其在量子距离着色参数中的应用，通过线性规划优化多项式选择以获得更强的界限。
The notion of duality is a key element in understanding the interplay between the stability and chromatic numbers of a graph. This notion is a central aspect in the celebrated theory of perfect graphs, and is further and deeply developed in the context of the Lovász theta function and its equivalent characterizations and variants. The main achievement of this paper is the introduction of a new family of norms, providing upper bounds for the stability number, that are obtained from duality from the norms motivated by Hoffman's lower bound for the chromatic number and which achieve the (compleme…
One of the best-known results in spectral graph theory is the inequality of Hoffman \[ χ\left( G\right) \geq1-\frac{λ\left( G\right) }{λ_{\min }\left( G\right) }, \] where $χ\left( G\right) $ is the chromatic number of a graph $G$ and $λ\left( G\right) ,$ $λ_{\min}\left( G\right) $ are the largest and the smallest eigenvalues of its adjacency matrix. In this note Hoffman's inequality is extended to weighted uniform $r$-graphs for every even $r$.
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
Spectral approaches for $d$-improper chromatic number
本文研究$d$-不适当着色的谱方法，加强Hoffman界并刻画等式情形，猜想色数与强积的$d$-不适当色数相等，并在完美图等特殊图类中证明。
In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a lower bound on the chromatic number of a pure $d$-dimensional simplicial complex in the terms of the spectra of the higher Laplacian operators.
A natural generalization of a regular (or equitable) partition of a graph, which makes sense also for non-regular graphs, is the so-called weight-regular partition, which gives to each vertex $u\in V$ a weight that equals the corresponding entry $ν_u$ of the Perron eigenvector $\mathbfν$. This paper contains three main results related to weight-regular partitions of a graph. The first is a characterization of weight-regular partitions in terms of double stochastic matrices. Inspired by a characterization of regular graphs by Hoffman, we also provide a new characterization of weight-regularity …
Efficient spectral bounds on the chromatic number of Hamming, Johnson, and Kneser graph powers
本文提出利用超几何正交多项式高效计算汉明图、约翰逊图和克奈塞尔图幂的谱界，从而在多项式时间内得出色数的下界。
We show that Hoffman's sum of eigenvalues bound for the chromatic number is at least as good as the Lovász theta number, but no better than the ceiling of the fractional chromatic number. In order to do so, we display an interesting connection between this sum of eigenvalues bound and a generalization of the Lovász theta number introduced by Manber and Narasimhan in 1988.
Hoffman proved that a graph $G$ with eigenvalues $μ_1 \ge \ldots \ge μ_n$ and chromatic number $χ(G)$ satisfies: \[ χ\ge 1 + κ\] where $κ$ is the smallest integer such that \[ μ_1 + \sum_{i=1}^κ μ_{n+1-i} \le 0. \] We strengthen this well known result by proving that $χ(G)$ can be replaced by the quantum chromatic number, $χ_q(G)$, where for all graphs $χ_q(G) \le χ(G)$ and for some graphs $χ_q(G)$ is significantly smaller than $χ(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves…
On the exact quantum chromatic number of generalized Johnson graphs
本文研究广义约翰逊图的量子色数，通过构造正交表示、结合特征值分析确定两类无穷族的精确量子色数，发现其经典色数随n指数增长、量子色数线性增长，存在指数级分离。
In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color. We compute the chromatic number of the root lattices, their duals, and of the Leech lattice, we consider the chromatic number of lattices of Voronoi's first kind, and we investigate the asymptotic behaviour of the chromatic number of lattices when the dimension tends to infinity. We introduce a spectral lower bound for the chromatic number of lattices in spir…
One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where mu_1 and mu_n are respectively the maximum and minimum eigenvalues of the adjacency matrix: chi >= 1 + mu_1 / (- mu_n). We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenvalues of the adjacency, Laplacian and signless Laplacian matrices. The various known bounds are also unified by considering the normalized adjacency matrix, and examples are cited for wh…
We consider several semidefinite programming relaxations for the max-$k$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-$k$-cut when $k>2$ that is applicable to any graph. This bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph. For regular graphs, the new bound on the chromatic number is the same as the well-known Hoffman …
We define the independence ratio and the chromatic number for bounded, self-adjoint operators on an L^2-space by extending the definitions for the adjacency matrix of finite graphs. In analogy to the Hoffman bounds for finite graphs, we give bounds for these parameters in terms of the numerical range of the operator. This provides a theoretical framework in which many packing and coloring problems for finite and infinite graphs can be conveniently studied with the help of harmonic analysis and convex optimization. The theory is applied to infinite geometric graphs on Euclidean space and on the…
The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\sc Hoffman} lower bound. Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the sums …
The exact quantum chromatic number of Hadamard graphs
本文利用Avis等人的上界和经Ganesan推广的类Hoffman下界，结合Ito关于共轭类图的结果，计算了特定阶数Hadamard图及其范畴积的精确量子色数。
Spectral Radius, Vertex Deletion, and Chromatic Number of Signed Graphs
研究符号图删除顶点后谱半径的变化，证明Sun-Das定理对符号图成立，并给出基于所有特征值的色数下界。
Hoffman proved that for a simple graph $G$, the chromatic number $χ(G)$ obeys $χ(G) \le 1 - \frac{λ_1}{λ_{n}}$ where $λ_1$ and $λ_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$ respectively. Lovász later showed that $χ(G) \le 1 - \frac{λ_1}{λ_{n}}$ for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lovász's theorem, then extend the technique to derive generalizations of Hoffman's theorem when allowed a certain proportion of edge-conflicts. Using this result, we show that if a 3-uniform hypergraph is 2-colorable, th…
Kemeny's constant via matrix compression and eigenvalue interlacing
本文通过矩阵压缩与特征值交错技术，为有限连通加权图的Kemeny常数建立谱下界，并给出色数相关的尖锐不等式及等式刻画，同时应用于归一化割与电导问题。
A graph energy conjecture through the lenses of semidefinite programming
本文推导图能量的半定规划表述，得到多个相关下界以推进Fajtlowicz图能量猜想的证明，还针对高度正则图类给出了基于Schrijver theta数的支持性界。
We introduce a generalization of the celebrated Lovász theta number of a graph to simplicial complexes of arbitrary dimension. Our generalization takes advantage of real simplicial cohomology theory, in particular combinatorial Laplacians, and provides a semidefinite programming upper bound of the independence number of a simplicial complex. We consider properties of the graph theta number such as the relationship to Hoffman's ratio bound and to the chromatic number and study how they extend to higher dimensions. Like in the case of graphs, the higher dimensional theta number can be extended t…
Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs
该研究针对非正则Borel pmp图，适配经典结果给出近似可测色数新下界、强化近似可测二部图刻画，还引入可测控制数并给出拉普拉斯算子谱隙界。
An edge-spectral Erdős-Stone-Simonovits theorem and its stability
本文证明无 F 图的谱半径平方与边数之间的统一上界，建立相应的边谱稳定性，并改进关于大谱半径图中公共邻点数量的最优下界。
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