合并A-和Q- spectral theories
Merging the A- and Q-spectral theories
摘要：
设$G$是一个具有邻接矩阵$A\left( G\right) $的图，$D\left( G\right) $是$G.$的度的对角矩阵，定义$G$的无符号拉普拉斯$Q\left( G\right) $为$Q\left( G\right) :=A\left( G\right) +D\left( G\right) $ . Cvetković把邻接矩阵的研究称为$A$ % \textit{-spectral theory}，把无符号拉普拉斯算子的研究称为$Q$ \textit{-spectral theory}。多年来，这两种理论之间的许多相似之处和差异已经建立。为了跟踪$A\left( G\right) $到$Q\left( G\right) $的渐变过程，本文提出了研究由\[ A_α\left( G\right) :=αD\left( G\right) +\left( 1-α\right) A\left( G\right) \text{, \ \ }0\leqα\leq1. \]定义的$A\left( G\right) $和$D\left( G\right) $的凸线性组合$A_{α}\left( G\right) $的方法。这项研究对$A\left( G\right) $和$Q\left( G\right) $有了新的认识，并得到了一些新的发现，特别是一个新的谱图兰定理。一些具有挑战性的开放问题进行了讨论。
英文摘要：
Let $G$ be a graph with adjacency matrix $A\left( G\right) $, and let $D\left( G\right) $ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right) $ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right) $. Cvetković called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right) $ into $Q\left( G\right) $ in this paper it is suggested to study the convex linear combinations $A_{α}\left( G\right) $ of $A\left( G\right) $ and $D\left( G\right) $ defined by \[ A_α\left( G\right) :=αD\left( G\right) +\left( 1-α\right) A\left( G\right) \text{, \ \ }0\leqα\leq1. \] This study sheds new light on $A\left( G\right) $ and $Q\left( G\right) $, and yields some surprises, in particular, a novel spectral Turán theorem. A number of challenging open problems are discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
