论文标题
给定独立数的最小光谱半径的图形
Graphs with the minimum spectral radius for given independence number
论文作者
论文摘要
令$ \ mathbb {g} _ {n，α} $为订单$ n $和独立数$α$的连接图。给定$ k = n-α$，在$ \ mathbb {g} _ {n，α} $之间具有最小光谱半径的图称为最小化图。 Stevanović在古典书中[D. Stevanović，《图形光谱半径》，学术出版社，阿姆斯特丹，2015年。]指出，在$ \ mathbb {g} _ {n，α} $中确定最小化图在$ 96 $上似乎是一个棘手的问题。最近，\ cite {lou}中的lou and guo证明了$ \ mathbb {g} _ {n，α} $的最小化图，如果$α\ ge \ ge \ lceil \ frac \ frac {n} {2} {2} {2} {2} \ rceil $。在本文中，我们进一步详细介绍了最小化图的结构特征，然后为其构建定理。因此，从理论上讲，我们完全确定$ \ mathbb {g} _ {n，α} $中的最小化图以及其光谱半径，对于任何给定的$ k = n-α\ le \ le \ frac {n} {2} $。作为一个应用程序，我们确定$ \ mathbb {g} _ {n，α} $中的所有最小化图，对于$α= n-1 = n-1，n-2，n-3，n-4，n-4，n-5，n-6 $以及它们的光谱半径，前四个结果在\ cite \ cite {xu，lou，lou，lou}和最后两个是新的。
Let $\mathbb{G}_{n,α}$ be the set of connected graphs with order $n$ and independence number $α$. Given $k=n-α$, the graph with minimum spectral radius among $\mathbb{G}_{n,α}$ is called the minimizer graph. Stevanović in the classical book [D. Stevanović, Spectral Radius of Graphs, Academic Press, Amsterdam, 2015.] pointed that determining minimizer graph in $\mathbb{G}_{n,α}$ appears to be a tough problem on page $96$. Very recently, Lou and Guo in \cite{Lou} proved that the minimizer graph of $\mathbb{G}_{n,α}$ must be a tree if $α\ge\lceil\frac{n}{2}\rceil$. In this paper, we further give the structural features for the minimizer graph in detail, and then provide of a constructing theorem for it. Thus, theoretically we completely determine the minimizer graphs in $\mathbb{G}_{n,α}$ along with their spectral radius for any given $k=n-α\le \frac{n}{2}$. As an application, we determine all the minimizer graphs in $\mathbb{G}_{n,α}$ for $α=n-1,n-2,n-3,n-4,n-5,n-6$ along with their spectral radii, the first four results are known in \cite{Xu,Lou} and the last two are new.
