On the Seidel spectrum of threshold graphs

关于由于超边和顶点删除导致的均匀超图的Seidel能量

On the Seidel energy of uniform hypergraphs due to hyperedge and vertex deletion

本文研究了完全3-均匀双部超图C^3_{m,n}的Seidel谱，证明其在单个超边删除后仍有一个负Seidel特征值，并证明在单个超边或顶点删除后其Seidel能量会减少。

Spectral symmetry in conference matrices

A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $Σ$ (that is, $Σ=-Σ$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.

On the eigenvalues of some signed graphs

Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1. Clearly, if $G$ is a graph of order $n$ with no isolated vertex, then the Seidel matrix of $G$ is also the adjacency matrix of a signed complete graph $K_n$ whose negative edges induce $G$. In this paper, we study the Seidel eigenvalues of the complete multipartite graph $K_{n_1,\ldots,n_k}$ and investigate its Seidel characteristic polynomial. We show that if ther…

超图的冠积的谱

The spectrum of the Corona of Hypergraphs

该研究推广图冠积运算提出超图冠积及广义冠积概念，推导正则超图冠积的邻接与Seidel谱、特征向量及相关矩阵性质，构造出无穷多组非正则非同构的同谱超图。

几乎所有图都由其广义谱确定

Almost all graphs are determined by their generalized spectrum

该研究证明了Haemers关于几乎所有图由广义谱确定的猜想，通过结合多种数学理论，证明随机图G(n,1/2)以趋于1的概率满足该性质。

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…

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.

On the Pebbling Threshold Spectrum

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. 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. In this note we show that the spectrum of pebbling thresholds for graph sequences spans the entire range from n^{1/2} to n. This answers a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remain…

自然图谱

Natural graph spectra

本文提出自然图矩阵的概念，证明存在一种矩阵其谱可确定随机图的同构类，并引入双重代数框架作为理论基础。

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。

The pebbling threshold spectrum and paths

Given a distribution of pebbles on the vertices of a graph, say that we can pebble a vertex if a pebble is left on it after some sequence of moves, each of which takes two pebbles from some vertex and places one on an adjacent vertex. A distribution is solvable if all vertices are pebblable; the pebbling threshold of a sequence of graphs is, roughly speaking, the total number of pebbles for which random distributions with that number of pebbles on a graph in the sequence change from being almost never solvable to being almost always solvable. We show that any sequence of connected graphs with …

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。

某些群上（真）幂图及其补图的通用邻接谱

Universal adjacency spectrum of (proper) power graphs and their complements on some groups

本文研究循环群、二面体群和广义四元数群上幂图、真幂图及其补图的通用邻接谱，确定若干情形的完整谱，并统一推广已有结果。

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.