arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 1707.04810math.CO
收藏

无长圈图的谱半径

The spectral radius of graphs without long cycles

Jun Gao, Xinmin Hou

备注：
17 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

Nikiforov证明了对于给定的整数$k\ge 2$，任何具有足够大阶$n$且谱半径为$μ(G)\geq μ(S_{n,k})$的图$G$，（或$μ(G)\ge μ(S_{n,k}^+))$包含$C_{2k+1}$或$C_{2k+2}$（或$C_{2k+2}$），除非$G=S_{n,k}$（或$G=S_{n,k}^+)$，其中$C_\ell$是长度为$\ell$和$S_{n,k}=K_k\vee \overline{K_{n-k}}$的圈，$k$阶的完全图和$n-k$顶点上的空图的连接图，并且$S_{n,k}^+$是通过在$S_{n,k}$的独立集合中添加边而从$S_{n,k}$获得的图。%这可以看作是Erdös和Sós猜想的谱版本。本文考虑Nikiforov猜想的一个较弱的版本，证明了对于给定的整数$k\ge 2$，任何具有谱半径$μ(G)\geq μ(S_{n,k})$的充分大阶$n$的图$G$（或$μ(G)\ge μ(S_{n,k}^+))$ % $C_{2k+1}$或$C_{2k+2}$（或$C_{2k+2}$），除非$G=S_{n,k}$（或$G=S_{n,k}^+)$ $S_{n,k}$（或$S_{n,k}^+$）是所有$n$阶图中具有最大半径的唯一极图，并且包含具有$\ell \geq 2k+1$的圈$C_{\ell}$（或具有$\ell \geq 2k+2$的$C_{\ell}$），除非$G=S_{n,k}$（或$G=S_{n,k}^+)$）。这些结果也暗示了Nikiforov在[Theorem 2，The谱半径of graphs without paths and cycles of specified length，LAA，2010]中给出的结果。

英文摘要：

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ (or $μ(G)\ge μ(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$, where $C_\ell$ is a cycle of length $\ell$ and $S_{n,k}=K_k\vee \overline{K_{n-k}}$, the join graph of a complete graph of order $k$ and an empty graph on $n-k$ vertices, and $S_{n,k}^+$ is the graph obtained from $S_{n,k}$ by adding an edge in the independent set of $S_{n,k}$. %This can be vie as spectral version of Erdös and Sós conjecture. In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $μ(G)\geq μ(S_{n,k})$ (or $μ(G)\ge μ(S_{n,k}^+))$ %$C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$$S_{n,k}$ ( or $S_{n,k}^+$) is the unique extremal graph with maximum radius among all of the graphs of order $n$ and contains a cycle $C_{\ell}$ with $\ell \geq 2k+1$ (or $C_{\ell}$ with $\ell \geq 2k+2$), unless $G=S_{n,k}$ (or $G=S_{n,k}^+)$. These results also imply a result of Nikiforov given in [Theorem 2, The spectral radius of graphs without paths and cycles of specified length, LAA, 2010].

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。

↑