Nankai University（南开大学）；Arizona State University（亚利桑那州立大学）
Shenwei Huang, Zilin Jiang
备注：
34 pages, 8 figures and 54 diagrams; revised version gives purely structural formulations of the classification theorems and adds ancillary Lean formalization
Guo and Royle recently classified the connected cubic graphs without eigenvalues of their adjacency matrix in the open interval $(-1, 1)$, and raised the question of extending their classification to graphs of maximum degree at most $3$. Together with their cubic classification, our result fully answers this question by characterizing all connected subcubic graphs that are not cubic and have no eigenvalues in $(-1,1)$. We show that exactly two infinite families and seven sporadic examples occur, and that every sporadic graph has at most $18$ vertices.
To obtain this complete classification, we build a bridge between spectral graph theory and structural graph theory for graphs whose adjacency matrices, after selected diagonal entries are changed to $-1$, have smallest eigenvalue at least $-2$. This generalizes the classical theorem of Cameron, Goethals, Seidel and Shult for graphs with smallest eigenvalue at least $-2$.
As a consequence, we prove that $(-1,1)$ is a maximal spectral gap set for the class of connected subcubic graphs. Guo and Royle, answering a question of Kollár and Sarnak, established this maximality for connected cubic graphs. Our result generalizes their conclusion to the subcubic setting.
证明了一个具有$m$边的三次图至少有$m/9$边的诱导匹配。我们的结果推广了Kang，Mnich和Müller关于可平面图的结果（Induced matching in subcubic可平面图，SIAM J. Discrete Math.26（2012）1383-1411），并解决了Henning和Rautenbach的猜想（Induced matching in subcubic graphs without short cycles，to appear in Discrete Math.）。
The $n/2$-bound for locating-dominating sets in subcubic graphs
Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS（克莱蒙奥弗涅大学，法国国家科学研究中心，圣艾蒂安矿业学院，克莱蒙奥弗涅国立理工学院，LIMOS）；Department of Mathematics and Applied Mathematics, University of Johannesburg（约翰内斯堡大学数学与应用数学系）；Department of Computer Science and Mathematics, Lebanese American University（黎巴嫩美国大学计算机科学与数学系）；Aix-Marseille Université, CNRS, École Centrale Méditerranée, LIS（艾克斯-马赛大学，法国国家科学研究中心，地中海中央理工学院，LIS）；Department of Computer Science and Engineering, SRM University-AP（SRM大学AP校区计算机科学与工程系）；Turku Collegium for Science, Medicine and Technology TCSMT, University of Turku（图尔库科学技术医学学院，图尔库大学）；Department of Mathematics and Statistics, University of Turku（图尔库大学数学与统计系）
Adjacency eigenvalues of graphs without short odd cycles
谱Turán型问题是图论中最经典的问题之一。本文考虑谱Turán型问题。设$G$是一个图，$\mathcal{G}$是一组图，如果$G$不包含$\mathcal{G}$的任何元素作为子图，我们称$G$是\textit{$\mathcal{G}$-free}.分别用$λ_1$和$λ_2$表示$G,$的邻接矩阵$A(G)$的最大和第二大特征值。本文根据图的邻接特征值刻画了不含短奇圈的图。首先，建立了$n$ -顶点$\{C_3,C_5,\ldots,C_{2k+1}\}$ -free图的$λ_1^{2k}+λ_2^{2k}$的上界，其中$k$是正整数.所有相应的极值图被识别。其次，给出了非二部图的谱半径包含长度至多为$2k+1$的奇圈的一个充分条件。最后，我们刻画了具有奇数围长至少为$2k+3,$的$n$ -顶点非二部图集合中唯一具有最大谱半径的图，解决了Lin，Ning和Wu [Eigenvalues and triangles in graphs，Combin.可能吧Comput. 30（2）（2021）258-270]。
