arXiv 2102.026742021-02-04math.CO

具有固定大小的图的谱极值：圈与完全二部图

Spectral extrema of graphs with fixed size: cycles and complete bipartite graphs

Mingqing Zhai, Huiqiu Lin, Jinlong Shu

备注：: 21 pages, 6 pages. This paper has been accepted for publication in European Journal of Combinatorics

摘要：

Nikiforov [图的最大特征值的一些不等式，Combin。可能吧Comput. 179- 189]表明，如果$G$不含$K_{r+1}$，则光谱半径为$ρ(G)\leq\sqrt{2m(1-1/r)}$，这意味着如果$ρ(G)>\sqrt{m}$，则$G$包含$C_3$。在本文中，我们遵循这个方向，确定哪些子图将包含在$G$，如果$ρ(G)> f(m)$，其中$f(m)\sim\sqrt{m}$作为$m\rightarrow \infty$。我们首先表明，如果$ρ(G)\geq \sqrt{m}$，那么$G$包含$K_{2,r+1}$，除非$G$是一个明星;和$G$包含$C_3^+$或$C_4^+$，除非$G$是一个完全的二部图，其中$C_t^+$表示从$C_t$和$C_3$通过识别边缘获得的图。其次，证明了若$ρ(G)\geq{\frac12+\sqrt{m-\frac34}}$，则$G$包含五边形和六边形，除非$G$是一本书;若$ρ(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$，则对每个$t\leq 2k+2$，$G$包含$C_t$.最后，本文还对相关的问题进行了讨论，为进一步的研究提供了参考。

英文摘要：

Nikiforov [Some inequalities for the largest eigenvalue of a graph, Combin. Probab. Comput. 179--189] showed that if $G$ is $K_{r+1}$-free then the spectral radius $ρ(G)\leq\sqrt{2m(1-1/r)}$, which implies that $G$ contains $C_3$ if $ρ(G)>\sqrt{m}$. In this paper, we follow this direction on determining which subgraphs will be contained in $G$ if $ρ(G)> f(m)$, where $f(m)\sim\sqrt{m}$ as $m\rightarrow \infty$. We first show that if $ρ(G)\geq \sqrt{m}$, then $G$ contains $K_{2,r+1}$ unless $G$ is a star; and $G$ contains either $C_3^+$ or $C_4^+$ unless $G$ is a complete bipartite graph, where $C_t^+$ denotes the graph obtained from $C_t$ and $C_3$ by identifying an edge. Secondly, we prove that if $ρ(G)\geq{\frac12+\sqrt{m-\frac34}}$, then $G$ contains pentagon and hexagon unless $G$ is a book; and if $ρ(G)>{\frac12(k-\frac12)+\sqrt{m+\frac14(k-\frac12)^2}}$, then $G$ contains $C_t$ for every $t\leq 2k+2$. In the end, some related conjectures are provided for further research.

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。

摘要：

英文摘要：

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