Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees

Let $R_T$ be the Ricci matrix of a finite tree $T$ introduced in \cite{BaiChengHua2026}, the largest eigenvalue ${\lambda }_{\max }(R_T)$ determines the sign of a discrete Einstein metric curvature on the tree. This paper investigates the asymptotic behavior of the sequence ${\lambda }_k={\lambda }_{\max }(R_{T_k})$ obtained by repeatedly adding pendant edges at a fixed vertex. We prove that ${\lambda }_k$ converges to a limit ${\lambda }_{\mathrm{\infty }}$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: $${\lambda }_k={\lambda }_{\mathrm{\infty }}+\frac{\alpha }{d+k}+O\text{\hspace{0.17em}⁣}\left(\frac{1}{(d+k{)}^2}\right),$$ where $d$ is the degree of the original vertex, and the coefficient $\alpha$ is given by a spectral projection. As a corollary, when $\alpha \mathrm{\ne }0$, ${\lambda }_k$ is eventually strictly monotonic (increasing or decreasing). This theory reveals the fine influence of local leaf addition on the global spectrum.