arXiv 2109.115462021-09-23math.CO

直径不超过4的子树存在的谱半径条件

Spectral radius conditions for the existence of all subtrees of diameter at most four

Xiangxiang Liu, Hajo Broersma, Ligong Wang

备注：: 16 pages
发表信息：: Linear Algebra and its Applications, 663 (2023), 80-101
DOI：: 10.1016/j.laa.2023.01.004

摘要：

用$μ(G)$表示图$G$的谱半径。我们部分证实了一个猜想，由于Nikiforov，这是一个谱半径模拟著名的Erdens-Sós猜想，任何树的顺序$t$是包含在一个图的平均度大于$t-2$。令$S_{n,k}=K_{k}\vee\overline{K_{n-k}}$，令$S_{n,k}^{+}$是从$S_{n,k}$通过添加连接$S_{n,k}$的独立集合的两个顶点的单个边而获得的图。在2010年，Nikiforov证明了对于给定的整数$k$，每一个具有足够大阶$n$和$μ(G)\geq μ(S_{n,k}^{+})$的图$G$包含所有阶为$2k+3$的树，除非$G=S_{n,k}^{+}$。我们证实了这一猜想的树木直径最多为4，有一个例外。事实上，我们证明了以下关于$k\geq 8$的更强结果。如果一个具有足够大阶数$n$的图$G$满足$μ(G)\geq μ(S_{n,k})$和$G\neq S_{n,k}$，则$G$包含所有阶数$2k+3$且直径至多为4的树，除了从一个星形$K_{1,k+1}$通过将其每个$k+1$边细分一次而得到的树。

英文摘要：

Let $μ(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erdős-Sós Conjecture that any tree of order $t$ is contained in a graph of average degree greater than $t-2$. Let $S_{n,k}=K_{k}\vee\overline{K_{n-k}}$, and let $S_{n,k}^{+}$ be the graph obtained from $S_{n,k}$ by adding a single edge joining two vertices of the independent set of $S_{n,k}$. In 2010, Nikiforov conjectured that for a given integer $k$, every graph $G$ of sufficiently large order $n$ with $μ(G)\geq μ(S_{n,k}^{+})$ contains all trees of order $2k+3$, unless $G=S_{n,k}^{+}$. We confirm this conjecture for trees with diameter at most four, with one exception. In fact, we prove the following stronger result for $k\geq 8$. If a graph $G$ with sufficiently large order $n$ satisfies $μ(G)\geq μ(S_{n,k})$ and $G\neq S_{n,k}$, then $G$ contains all trees of order $2k+3$ with diameter at most four, except for the tree obtained from a star $K_{1,k+1}$ by subdividing each of its $k+1$ edges once.

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