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In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \\, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \\leq 2, \\; m_{G}[d_{n-1}, 2n-2] = 1, \\; m_{G}[0, d_1] \\le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. In addition, we obtain all disconnected graphs such that $m_{G}[0, d_1] = 3"},"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":"2605.27405","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2026-05-04T19:40:31Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"5c605591677b757e06ac5fe4a48ebe30905fe000beefeab3ad2bb6805947a4bc","abstract_canon_sha256":"9fddf3711d6084493b6f77cbe1c48ef4d819d079882e49deade29a3cbc671abe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T00:05:17.148576Z","signature_b64":"gRm4vh2SZgUKsdwN4MWN5KAWBgChgl73Xs+253tB+bAaPHdRr52qH20Tm5xvlVIAxH1CNcXxtfNg0M5i40/mAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf0710acd20ca0d344c8a15d0c2ec7ff0d6faa661c71ca74cc9dfaa469b9a7fb","last_reissued_at":"2026-05-28T00:05:17.147887Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T00:05:17.147887Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distribution of signless Laplacian eigenvalues and degree sequence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.CO","authors_text":"D. Tracina, L. S. de Lima, M. Darougheh, Saieed Akbari","submitted_at":"2026-05-04T19:40:31Z","abstract_excerpt":"Let $G$ be a graph of order $n$ with degree sequence $d_1 \\geq \\cdots \\geq d_{n}$. Let $m_{G}I$ be the number of signless Laplacian eigenvalues in an interval $I$. In this paper, we characterize the distribution of the signless Laplacian eigenvalues in terms of the degree sequence of a graph within specific subintervals of $[0, \\, 2n-2].$ We determine all graphs $G$ such that $m_{G}[d_n, 2n-2] \\leq 2, \\; m_{G}[d_{n-1}, 2n-2] = 1, \\; m_{G}[0, d_1] \\le 2.$ We also prove that there is no graph such that $m_{G}[0, d_3]=1$. 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