Home > ELA > Vol. 31 (2016)

#### Keywords

Maximum degree, Optimizing, Spectral radius, Tree

#### Abstract

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

#### Recommended Citation

Du, Xue and Shi, Lingsheng.
(2016),
"Trees with given maximum degree minimizing the spectral radius",
*Electronic Journal of Linear Algebra*,
Volume 31, pp. 335-361.

DOI: http://dx.doi.org/10.13001/1081-3810.3323

*Abstract*