{"paper":{"title":"On the walk matrix of the Dynkin graph $D_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chuanming Wang, Songlin Guo, Wei Wang","submitted_at":"2022-02-27T03:20:46Z","abstract_excerpt":"Let $W(D_n)$ denote the walk matrix of the Dynkin graph $D_n$, a tree obtained from the path of order $n-1$ by adding a pendant edge at the second vertex. We prove that $\\text{rank}\\,W(D_n)=n-2$ if $4\\mid n$ and $\\text{rank}\\,W(D_n)=n-1$ otherwise. Furthermore, we prove that the Smith normal form of $W(D_n)$ is $$\\text{diag}[\\underbrace{1,1,\\ldots,1}_{\\lceil\\frac{n}{2}\\rceil},\\underbrace{2,2,\\ldots,2}_{\\lfloor\\frac{n}{2}\\rfloor-1},0]$$ when $4\\nmid n$. This confirms a recent conjecture in [W.Wang, F.Liu, W.Wang, Generalized spectral characterizations of almost controllable graphs, European J. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.13279","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/2202.13279/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"}