阈值图的Colin de Verdière图参数

超稳定张拉整体结构与 Colin de Verdière 数 $ν$

Super Stable Tensegrities and the Colin de Verdière Number $ν$

本文建立了多重图可实现为超稳定张拉整体的最大维数与谱图论中Colin de Verdière数ν的精确关系，并由此给出三维超稳定张拉整体多重图的组合刻画。

A graph minors characterization of signed graphs whose signed Colin de Verdière parameter $ν$ is two

A signed graph is a pair $(G,Σ)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,...,n\}$ and $Σ\subseteq E$. The edges in $Σ$ are called odd and the other edges even. By $S(G,Σ)$ we denote the set of all symmetric $n\times n$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}\in \mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i\not=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} \in \mathbb{R}$ for …

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。

Even maps, the Colin de~Verdière number and representations of graphs

Van der Holst and Pendavingh introduced a graph parameter $σ$, which coincides with the more famous Colin de Verdière graph parameter $μ$ for small values. However, the definition of $σ$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $μ(G) \leq σ(G) + 2$ and conjectured $μ(G) \leq σ(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $μ(G)$ which is, in general, tight. Equality between $μ$ and $σ$ does not hold in general as van der Holst and Pendavingh showed that there is…

源于非分离平面图的构造及其 Colin de Verdière 不变量

Constructions stemming from non-separating planar graphs and their Colin de Verdière invariant

本文利用极大非分离平面图构造极大无链图与无结图，证明了其补图的 Colin de Verdière 不变量取值，并验证了相关猜想，得出高阶非分离平面图补图本质有链和有结的结论。

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 $v$.

The Strong Arnold Property for 4-connected flat graphs

We show that if $G=(V,E)$ is a 4-connected flat graph, then any real symmetric $V\times V$ matrix $M$ with exactly one negative eigenvalue and satisfying, for any two distinct vertices $i$ and $j$, $M_{ij}<0$ if $i$ and $j$ are adjacent, and $M_{ij}=0$ if $i$ and $j$ are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric $V\times V$ matrix $X$ with $MX=0$ and $X_{ij}=0$ whenever $i$ and $j$ are equal or adjacent. (A graph $G$ is {\em flat} if it can be embedded injectively in $3$-dimensional Euclidean space such that the image of any circuit is the boundary of some…

Techniques for determining equality of the maximum nullity and the zero forcing number of a graph

It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters which bound the maximum nullity of a graph below. In particular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph paramete…

Using a new zero forcing process to guarantee the Strong Arnold Property

The maximum nullity $M(G)$ and the Colin de Verdière type parameter $ξ(G)$ both consider the largest possible nullity over matrices in $\mathcal{S}(G)$, which is the family of real symmetric matrices whose $i,j$-entry, $i\neq j$, is nonzero if $i$ is adjacent to $j$, and zero otherwise; however, $ξ(G)$ restricts to those matrices $A$ in $\mathcal{S}(G)$ with the Strong Arnold Property, which means $X=O$ is the only symmetric matrix that satisfies $A\circ X=O$, $I\circ X=O$, and $AX=O$. This paper introduces zero forcing parameters $Z_{\mathrm{SAP}}(G)$ and $Z_{\mathrm{vc}}(G)$, and proves that…

The Signed Positive Semidefinite Matrix Completion Problem for Odd-$K_4$ Minor Free Signed Graphs

We give a signed generalization of Laurent's theorem that characterizes feasible positive semidefinite matrix completion problems in terms of metric polytopes. Based on this result, we give a characterization of the maximum rank completions of the signed positive semidefinite matrix completion problem for odd-$K_4$ minor free signed graphs. The analysis can also be used to bound the minimum rank over the completions and to characterize uniquely solvable completion problems for odd-$K_4$ minor free signed graphs. As a corollary we derive a characterization of the universal rigidity of odd-$K_4$…