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We prove that if $G$ has an odd number of vertices and $\\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\\lambda\\equiv2\\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\\lambda\\equiv0\\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs"},"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":"1201.3221","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-16T11:36:47Z","cross_cats_sorted":[],"title_canon_sha256":"472a00bde673a211be3b39d865d0df244ffcd1cafcb86220ccc76d9e65bd4428","abstract_canon_sha256":"c80de11950ae477e583e71dbf519231fc717dc8270bf9371d9dc559e6cbec622"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:50.019782Z","signature_b64":"zsQYx6Fb0fi46IUsTo3yv3V9ZipeCv1iTUvDES+gtvIHt5P9wtA9fOCm+53MM4z+/0ZrbWgWSPq75AnS86bgDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6745557b5838fd7e388ed2c11e6c6fb78e322085ad085777be790039ec887d21","last_reissued_at":"2026-05-18T03:00:50.019115Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:50.019115Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spanning trees and even integer eigenvalues of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ebrahim Ghorbani","submitted_at":"2012-01-16T11:36:47Z","abstract_excerpt":"For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\\lambda\\equiv2\\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\\lambda\\equiv0\\pmod4$ and such an eigenvalue is simple. 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