图的反邻接矩阵是由图的距离矩阵通过保持每一行和每一列的最大距离而构造的。这个矩阵可以被解释为邻接矩阵的反面，而邻接矩阵是通过在每行和每列中只保持距离等于1而从图的距离矩阵构造的。图的（反）邻接特征值是它的（反）邻接矩阵的特征值。利用Haemers [Spectral characterization of mixed extensions of small graphs，Discrete Math.342（2019）2760--2764]提出的一种新技术，我们刻画了所有连通图恰好具有一个正的反邻接特征值，这类似于Smith的经典结果，即连通图恰好具有一个正的邻接特征值当且仅当它是完全多部图。在此基础上，我们确定了除了最多两个反邻接特征值等于$-2$和$0$的连通图。此外，对于反邻接矩阵，我们确定的HL-指数的图正好有一个正的反邻接特征值，其中的HL-指数测量多大的绝对值可能是一个图的中值特征值。最后提出了有待进一步研究的问题。
英文摘要：
The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers [Spectral characterization of mixed extensions of small graphs, Discrete Math. 342 (2019) 2760--2764], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph with exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to $-2$ and $0$. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.
Adjacency labelling for proper minor-closed graph classes
University of Ottawa（渥太华大学）；LaBRI, University of Bordeaux（波尔多大学 LaBRI）；Université libre de Bruxelles（布鲁塞尔自由大学）；Jagiellonian University（雅盖隆大学）；Carleton University（卡尔顿大学）；Monash University（蒙纳士大学）
The number of ideals of $\mathbb{Z}[x]$ containing $x(x-α)(x-β)$ with given index
众所周知，一个连通正则图是强正则的当且仅当它的邻接矩阵恰好有三个特征值。令$B$表示整数方阵，$\langle B \rangle$表示由$B$生成的全矩阵环的子环。则$\langle B \rangle$是有限秩的自由$\mathbb{Z}$ -模，它保证了$\langle B \rangle$在给定的有限指标下只存在1/2个理想.因此，形式狄利克雷级数$ζ_{\langle B \rangle}(s)=\sum_{n\geq 1}a_n n^{-s}$是定义良好的，其中$a_n$是$\langle B \rangle$的具有指标$n$的理想数。在这篇文章中，我们的目标是找到一个明确的形式，$ζ_{\langle B \rangle}(s)$时，$B$正好有三个特征值，所有这些都是整数，例如，强正则图的邻接矩阵，它不是顶点数非平方的会议图。根据环的同构定理，$\langle B \rangle$同构于$\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$，其中$m(x)$是$B$在$\mathbb{Q}$上的最小多项式，对于每个$γ\in \mathbb{Z}$，$\mathbb{Z}[x]/m(x)\mathbb{Z}[x]$同构于$\mathbb{Z}[x]/m(x+γ)\mathbb{Z}[x]$.因此，问题归结为计算具有给定有限指数的$\mathbb{Z}[x…
Constructions of regular sparse anti-magic squares
图的标号是图论中一个著名的研究热点问题。稀疏反幻方在构造图的顶点魔术标号中是很有用的。对于正整数$n,d$和$d<n$，基于$\{0,1,\cdots,nd\}$的$n\times n$数组$A$称为\emph{a sparse anti-magic square of order $n$ with density $d$}，用SAMS $(n,d)$表示，如果$\{1,2,\cdots,nd\}$的每个元素恰好出现$A$的一个条目，并且其行和、列和两个主对角线和构成一组$2n+2$连续整数。如果在每行、每列和每个主对角线中恰好有$d$正条目，则SAMS $(n,d)$被称为\emph{regular}。本文研究了$n\equiv1,5\pmod 6$阶正则稀疏反幻方的存在性，证明了对任意$n\equiv1,5\pmod 6$，存在正则SAMS $(n,d)$当且仅当$2\leq d\leq n-1$ .
Homological spectral graph theory and weighted cycle counting
School of Informatics, Xiamen University（厦门大学信息学院）；Department of Mathematics and Statistics, University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校数学与统计系）
