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It is a conjecture on Laplacian spread of graphs that $\\mu_1(G)-\\mu_{n-1}(G)\\le n-1$ or equivalently $\\mu_1(G)+\\mu_1(\\Gb)\\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $\\mu_1(G)\\mu_1(\\Gb)\\le n(n-1)$. Aouchiche and Hansen [A survey of Nordhaus--Gaddum type relations, Discrete Appl. Math. 161 (2013), 466--546] conjectured that %for any graph $G$ with"},"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":"1402.2995","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-02-12T22:08:12Z","cross_cats_sorted":[],"title_canon_sha256":"0b08eb751fabd778557355bbaf465db1a23b81139104c5a495686a2c1a83d3db","abstract_canon_sha256":"c33f183c4b23caa4424c5c6f0a6757dd23ebd861a9f7b78d3dab7207499ebbe9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:59:08.099327Z","signature_b64":"ifNyEVxC3hYQ1SJTLlZXHLIoUPyjZEzkVq+RnPuXcTYiNoVX42Ot/+QnX8K9Rq2+XpOUPq2LmjO8L+BUv+bhDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"48954253fbcc10f6ce7f0780eb5cb796359b37f0fa984d906f5d874b55acbc3b","last_reissued_at":"2026-05-18T02:59:08.098480Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:59:08.098480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nordhaus--Gaddum type inequalities for Laplacian and signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"B. 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