{"paper":{"title":"Spectral Monotonicity under Leaf Attachment and Limiting Behavior in Discrete Einstein Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.DG","authors_text":"Bobo Hua, Haoxuan Cheng, Shuliang Bai","submitted_at":"2026-05-22T08:47:33Z","abstract_excerpt":"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 $\\lambda_\\infty$ that depends only on the local branch data of $T$, and establish a first-order asymptotic expansion: \\[ \\lambda_k = \\lambda_\\infty + \\frac{\\alpha}{d+k} + O\\!\\left(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23379","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.23379/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}