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We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin.\n  Our main result shows that, whenever $d\\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. We further construct an infinite"},"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":"2607.05472","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T08:53:37Z","cross_cats_sorted":[],"title_canon_sha256":"b87aaf0411c71b7e074c41d365571d02504f4e7a5298cf1651d3938bc9b29183","abstract_canon_sha256":"b31a666377629de90685a38bd5c5f214d5466b5f85618b64c80c7d609af44964"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-08T01:18:11.207138Z","signature_b64":"Ddh3qyBhI+FcBp2KqYvofSCnvpCz9UM+VG7QxEaH32ToDSkFrRmjTY21t0pFZhrdWfRgpHWX4zB3N5da13c5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e07ffb4d9d77d5ee10e108e79662db7000b4e25a690b6e112edc7dd7dc1c29e","last_reissued_at":"2026-07-08T01:18:11.206712Z","signature_status":"signed_v1","first_computed_at":"2026-07-08T01:18:11.206712Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Second Largest Eigenvalue of Stiffness Matrices of Normalized Complete Frameworks","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lu Lu, Tingting Wang","submitted_at":"2026-07-06T08:53:37Z","abstract_excerpt":"Let $R(G,p)$ be the normalized rigidity matrix of a framework $(G,p)$ in $\\mathbb R^d$, and let \\[ L(G,p)=R(G,p)R(G,p)^{T} \\] be the associated stiffness matrix. We study the extremal eigenvalues of $L(K_n,p)$ for complete frameworks whose vertices lie on the unit sphere and have centroid at the origin.\n  Our main result shows that, whenever $d\\ge2$ and the image of $p$ contains at least three distinct points, the second largest eigenvalue of $L(K_n,p)$ is exactly $n/2$. This settles the eigenvalue part of a conjecture of Lew et al. [Israel J. Math. 256, 2023]. 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