AI 中文总结
本文研究了平衡树悬吊图的谱Turán问题，确定了在特定条件下谱极值族与普通极值族的不相交性。
AI 中文摘要
谱Turán理论中的核心问题是理解谱极值族SPEX(n,F)与普通极值族EX(n,F)之间的关系。对于许多禁止图F，已知在无限多个n的情况下，SPEX(n,F)⊆EX(n,F)成立，而只有少数例子显示两个族是不相交的。本文研究了平衡树悬吊图的情况。如果树的两个二分图类大小差异不超过1，则称为平衡树。设T是一个有2k或2k+1个顶点的平衡树，\widehat{T}是其悬吊图，通过在T的每个顶点添加一个新顶点得到。我们的第一个主要结果在T满足某些温和条件的情况下，为悬吊图\widehat{T}的谱Turán数建立了紧上界。我们的第二个结果确定了对于哪些整数k和非路径平衡树T，在2k或2k+1个顶点的情况下，存在无限多个整数n使得EX(n,\widehat{T})∩SPEX(n,\widehat{T})=∅。
英文摘要
A central problem in spectral Turán theory is to understand the relationship between the spectral extremal family ${\rm SPEX}(n,F)$ and the ordinary extremal family ${\rm EX}(n,F)$. For many forbidden graphs $F$, it is known that ${\rm SPEX}(n,F)\subseteq{\rm EX}(n,F)$ holds for infinitely many $n$, while only a few examples have been identified where the two families are disjoint. In this paper, we study this problem for suspensions of balanced trees. A tree is balanced if its two bipartition classes differ in size by at most one. Let $T$ be a balanced tree on $2k$ or $2k+1$ vertices and $\widehat T$ be its suspension which is obtained from $T$ by adding one new vertex adjacent to every vertex of $T$. Our first main result establishes a tight upper bound for the spectral Turán number of $\widehat T$ for sufficiently large $n$ provided that $T$ satisfies some mild assumptions. Our second result determines for which integers $k$ and which non-path balanced trees $T$ on $2k$ or $2k+1$ vertices there are infinitely many integers $n$ such that ${\rm EX}(n,\widehat{T})\cap {\rm SPEX}(n,\widehat{T})=\emptyset$.
Comments19 pages
