多面体体积比、Izmestiev 的 Colin de Verdiere 矩阵与谱隙

The Colin de Verdière parameter, excluded minors, and the spectral radius

In this paper we characterize graphs which maximize the spectral radius of their adjacency matrix over all graphs of Colin de Verdière parameter at most $m$. We also characterize graphs of maximum spectral radius with no $H$ as a minor when $H$ is either $K_r$ or $K_{s,t}$. Interestingly, the extremal graphs match those which maximize the number of edges over all graphs with no $H$ as a minor when $r$ and $s$ are small, but not when they are larger.

On the Colin de Verdiere graph number and penny graphs

The Colin de Verdiere number of graph G, denoted by μ(G), is a spectral invariant of G that is related to some of its topological properties. For example, μ(G) \leq 3 iff G is planar. A penny graph is the contact graph of equal-radii disks with disjoint interiors in the plane. In this note we prove lower bounds on μ(G) when the complement \bar{G} is a penny graph.

Colin de Verdière数与多面体图

The Colin de Verdière number and graphs of polytopes

图$G$的柯林·德·维迪埃数$μ(G)$是柯林·德·维迪埃矩阵对于$G$的最大余秩（即，在$G$上具有单个负特征值的薛定谔算子的最大余秩）。在2001年，Lovász给出了一个构造，它将每个凸3-多面体与其1-骨架的corank为3的Colin de Verdière矩阵相关联。我们将Lovász构造推广到更高的维度，将其解释为减去极对偶的体积的黑塞矩阵。作为推论，如果$G$是凸$d$ -多面体的1-骨架，则为$μ(G) \ge d$。确定签名的海森的体积是基于第二闵可夫斯基不等式的混合体积和波尔的条件平等。

关于 Colin de Verdière 谱图参数的三点观察

Three observations on the Colin de Verdière spectral graph parameter

本文围绕 Colin de Verdière 参数 μ 给出三点结果：否定 CdV 矩阵 Perron–Frobenius 特征向量相关问题，补充横截性自动成立的情形，并用计算机验证10维环面等曲面上界猜想失效。

Expected values of parameters associated with the minimum rank of a graph

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdière-type parameters. Let $G(v,p)$ denote the usual Erdős-Rényi random graph on $v$ vertices with edge probability $p$. We obtain bounds for the expected value of the random variables ${\rm mr}(G(v,p))$, ${\rm M}(G(v,p))$, $ν(G(v,p))$ and $ξ(G(v,p))$, which yield bounds on the average values of these parameters over all labeled graphs of order $v$.

Generalizations of the Strong Arnold Property and the minimum number of distinct eigenvalues of a graph

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a better understanding of all possible spectra and their associated multiplicities. These new properties are referred to as the Strong Spectral Property and the Strong Multiplicity Property. Finally, these ideas are applied to the minimum number of dist…