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Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.05117","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-17T11:32:24Z","cross_cats_sorted":[],"title_canon_sha256":"705dda62bfc28e9f0a0db87d8f8ee5fcf09093c6e31fd81e729f64b758dedffe","abstract_canon_sha256":"7124bed7777e610947fa3bacc48d87ec881e4711315fa92195a5a6acc81a6a04"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:32.830238Z","signature_b64":"lTOATY9tF+Sd32o0G1UoIsemd/hi/kpOqMhxyMc9XfItwfiAuxTw3y7w1fgoWr76mHAKJDmWG1O/lJyJ2zUAAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09498d134845d0a78bb313c204527602c3e07579a64e1dd36ebe4361a5087b0c","last_reissued_at":"2026-05-18T01:17:32.829626Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:32.829626Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a conjecture on `plateaux' phenomenon of graph Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ebrahim Ghorbani","submitted_at":"2015-10-17T11:32:24Z","abstract_excerpt":"Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. 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