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

Abstract

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