On the bounds of Laplacian eigenvalues of $k$-connected graphs

Czechoslovak Mathematical Journal

Yaoping Hou¹  Xiaodan Chen² 

[1] Department of Mathematics, Hunan Normal University, No. 15, Taozihu Road, Changsha, Hunan, P. R. China;Department of Mathematics, Hunan Normal University, No. 15, Taozihu Road, Changsha, Hunan, P. R. China, and College of Mathematics and Information Science, Guangxi University, No. 100, Daxue Road, Nanning, Guangxi, P. R. China

关键词: $k$-connected graph;    non-regular graph;    algebraic connectivity;    Laplacian spectral radius;    maximum degree;   

DOI  :  

学科分类：数学（综合）

来源: Akademie Ved Ceske Republiky

【 摘 要 】

Let $\mu_{n-1}(G)$ be the algebraic connectivity, and let $\mu_1(G)$ be the Laplacian spectral radius of a $k$-connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that \begin{equation*} \mu_{n-1}(G)\geq\frac{2nk^2}{(n(n-1)-2m)(n+k-2)+2k^2}, \end{equation*} with equality if and only if $G$ is the complete graph $K_n$ or $K_n-e$. Moreover, if $G$ is non-regular, then \begin{equation*} \mu_1(G)<2\Delta-\frac{2(n\Delta-2m)k^2}{2(n\Delta-2m)(n^2-2n+2k)+nk^2}, \end{equation*} where $\Delta$ stands for the maximum degree of $G$. Remark that in some cases, these two inequalities improve some previously known results.

【 授权许可】

Unknown