On the Pebbling Threshold Spectrum

The Pebbling Threshold of the Square of Cliques

Given an initial configuration of pebbles on a graph, one can move pebbles in pairs along edges, at the cost of one of the pebbles moved, with the objective of reaching a specified target vertex. The pebbling number of a graph is the minimum number of pebbles so that every configuration of that many pebbles can reach any chosen target. The pebbling threshold of a sequence of graphs is roughly the number of pebbles so that almost every (resp. almost no) configuration of asymptotically more (resp. fewer) pebbles can reach any chosen target. In this paper we find the pebbling threshold of the seq…

Pebbling in Dense Graphs

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertic…

网格上的石子游戏阈值

Thresholds for Pebbling on Grids

本文确定了网格上石子游戏的弱阈值和路径上的强阈值，引入新的中心性和权重引理，改进已有结果，并给出反例否定Graham猜想的阈值版本。

A Note on Graph Pebbling

We say that a graph G is Class 0 if its pebbling number is exactly equal to its number of vertices. For a positive integer d, let k(d) denote the least positive integer so that every graph G with diameter at most d and connectivity at least k(d) is Class 0. The existence of the function k was conjectured by Clarke, Hochberg and Hurlbert, who showed that if the function k exists, then it must satisfy k(d)=Ω(2^d/d). In this note, we show that k exists and satisfies k(d)=O(2^{2d}). We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property.

阿贝尔群中具有小交叉数的零和序列的阈值

Thresholds for zero-sums with small cross numbers in abelian groups

本文利用图 pebbling 的概率定理，为特定形式的阿贝尔群证明了 Geroldinger 定理的阈值版本，即当序列长度超过阈值时，零和序列为好的概率趋于 1。

几乎堆叠猜想：Sjöstrand覆盖卵石定理的一个推测性类比

The Almost Stacked Hypothesis: A Conjectural Analogue of Sjöstrand's Cover Pebbling Theorem

本文提出几乎堆叠猜想（ASH），证明其对偶圈和树的堆叠与清除数成立，并构造边密度趋于零的完美可卵石图无限族，表明高边密度并非必要条件。

On the normalized spectrum of threshold graphs

In this article we investigate normalized adjacency eigenvalues (simply normalized eigenvalues) and normalized adjacency energy of connected threshold graphs. A threshold graph can always be represented as a unique binary string. Certain eigenvalues are obtained directly from its binary representation and the rest of the eigenvalues are evaluated from its normalized equitable partition matrix. Finally, we characterize threshold graphs with at most five distinct eigenvalues.

关于爆破阈值的谱和结构

On the spectrum and structure of blowup thresholds

研究爆破阈值如何迫使足够密集的最大H-自由图成为有限图的爆破，揭示其与色数阈值的区别和非单调性。

On the Seidel spectrum of threshold graphs

In this paper, we analyse spectral properties of Seidel matrix (denoted by $S$) of connected threshold graphs. We compute the characteristic polynomial and determinant of Seidel matrix of threshold graphs. We derive formulas for the multiplicity of the eigenvalues $\pm 1$ of $S$. Further we determine threshold graphs with at most 5 distinct Seidel eigenvalues. Finally we construct families of Seidel cospectral threshold graphs.

树的谱游走判定的一个严格匹配数阈值

A Sharp Matching-Number Threshold for Spectral-Walk Determination of Trees

本文确定树的谱游走判定的匹配数阈值：匹配数≤4的树可由邻接谱与3≤k≤8的总游走序列唯一确定，匹配数为5时存在反例，证明基于有限核约化与代数重构。

Anti-regular graphs with loops and their spectrum

We characterize the graphs with loops whose degree sequences have no repeated values and find their adjacency spectrum. In the case of simple graphs, such graphs are called anti-regular graphs and are examples of threshold graphs. The spectrum is found by using the known relationship between tridiagonal matrices and orthogonal polynomials.

Spectrum graph coloring and applications to WiFi channel assignment

We introduce and explore a family of vertex-coloring problems which, surprisingly enough, have not been considered before despite stemming from the problem of Wi-Fi channel assignment. Given a spectrum of colors, endowed with a matrix of interferences between each pair of colors, the Threshold Spectrum Coloring problem fixes the number of colors available and aims to minimize the interference threshold, i.e., the maximum of the interferences at the vertices. Conversely, the Chromatic Spectrum Coloring problem fixes a threshold and aims to minimize the number of colors for which respecting that…