On the extremal connective eccentricity index among trees with maximum degree

The connective eccentricity index (CEI) of a graph $G$ is defined as $xi^{ce}(G)=sum_{v in V(G)}frac{d_G(v)}{varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique trees with the maximum and minimum CEI among all $n$-vertex trees and $n$-vertex conjugated trees with fixed maximum degree, respectively.