设$\mathcal{F}$是一个图族, 如果$G$不包含$\mathcal{F}$中的任意一个图作为子图, 则称$G$是禁用图族$\mathcal{F}$的。禁用图族$\mathcal{F}$的$n$阶图所能达到的最大边数称为$\mathcal{F}$的Turán数, 记作$ex(n, \mathcal{F})$。线性森林是指连通分支都是路或孤立点的图。设$\mathcal{L}_{n, k}'$是一个除图$P_{k+1}\cup(n-k-1)K_1$外, 包含其他所有边数为$k$的$n$阶线性森林组成的图族。本文给出了$\mathcal{L}_{n, k}'$的Turán数和对应的极值图。进一步我们给出了禁用$\mathcal{L}_{n, k}'$的$n$阶图的谱半径的最大值和对应的极值图。

Let $\mathcal{F}$ be a family of graphs. If $G$ does not contain every graph in $\mathcal{F}$ as a subgraph, then $G$ is said to be $\mathcal{F}$-free. The Turán number $ex(n, \mathcal{F})$ is defined to be the maximum number of edges in a graph on $n$ vertices that is $\mathcal{F}$-free. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n, k}'$ be the family of all linear forests of order $n$ with $k$ edges except $P_{k+1}\cup(n-k-1)K_1$. In this paper, we give Turán number of $\mathcal{L}_{n, k}'$ and corresponding extremal graphs. Furthermore, we give the maximum spectral radius of graphs of order $n$ that are $\mathcal{L}_{n, k}'$-free and the corresponding extremal graphs.