论文标题
关于光谱极端问题的猜想
On a conjecture of spectral extremal problems
论文作者
论文摘要
对于简单的图形$ f $，让$ \ mathrm {ex}（n，f）$和$ \ mathrm {ex_ {sp}}（n，n，f）$表示最大边数的图集和最大值的图集和带有最大频谱的最大频谱radius在$ n $ vertex中，无需任何图形$ f $ f $ f $ f $ f $。 Turán图$ t_ {n，r} $是$ n $ dertices上的完整$ r $ - 分段图，其零件大小尽可能相等。 Cioabă，Desai和Tait [没有奇数车轮的图形频谱半径，欧洲J. Combin。，99（2022）103420]提出了以下猜想：让$ f $是任何图形，以至于这些图是$ \ mathrm {ex}（ex ex}（ex ex}（n，f）$的图形，然后$ \ mathrm {ex_ {sp}}（n，f）\ subset \ mathrm {ex}（n，f）$，用于足够大的$ n $。在本文中，我们考虑图$ f $，以便通过添加$ o（1）$ edges从$ t_ {n，r} $获得$ \ mathrm {ex}（ex}（n，f）$对于$ n $，足够大。然后，Cioabă，Desai和Tait的猜想得到了完全解决。
For a simple graph $F$, let $\mathrm{Ex}(n, F)$ and $\mathrm{Ex_{sp}}(n,F)$ denote the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph $F$, respectively. The Turán graph $T_{n,r}$ is the complete $r$-partite graph on $n$ vertices where its part sizes are as equal as possible. Cioabă, Desai and Tait [The spectral radius of graphs with no odd wheels, European J. Combin., 99 (2022) 103420] posed the following conjecture: Let $F$ be any graph such that the graphs in $\mathrm{Ex}(n,F)$ are Turán graphs plus $O(1)$ edges. Then $\mathrm{Ex_{sp}}(n,F)\subset \mathrm{Ex}(n,F)$ for sufficiently large $n$. In this paper we consider the graph $F$ such that the graphs in $\mathrm{Ex}(n, F)$ are obtained from $T_{n,r}$ by adding $O(1)$ edges, and prove that if $G$ has the maximum spectral radius among all $n$-vertex graphs not containing $F$, then $G$ is a member of $\mathrm{Ex}(n, F)$ for $n$ large enough. Then Cioabă, Desai and Tait's conjecture is completely solved.
