The ɛ-spectral radius of trees with perfect matchings

Abstract

Let $G$ be a connected graph. The eccentricity matrix of $G$, denoted by $\varepsilon \left(G\right)$, is constructed from the distance matrix $D\left(G\right)$ by retaining the largest distances in each row and each column, and setting the remaining entries as 0. The $\varepsilon$-spectral radius of $G$ is the largest eigenvalue of $\varepsilon \left(G\right)$. In this paper, we determine the trees having the minimum $\varepsilon$-spectral radius among $2n$-vertex trees with perfect matchings.