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Here, we consider the Grone's inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin. Multilin. Alg. 39 (1995) 133--136] $$ \\sum_{i=1}^{k} \\mu_{i}(G) \\geq \\sum_{i=1}^{k} d_{i}(G)+1$$ and prove that for $k=2$, the equality holds if and only if $G$ is the star graph $S_{n}.$ The signless Laplacian version of Grone'"},"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":"1412.0323","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2014-12-01T01:23:02Z","cross_cats_sorted":[],"title_canon_sha256":"a3d8139c416252dab6f1e8c1eb82fec2aaa309a0af6b7a7716d2d87fe48a0506","abstract_canon_sha256":"c186311f666f197f6c413ac28a421cf6399ca40749e8f3961169892b3099d0be"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:28.082055Z","signature_b64":"hhBT2rJYr4l2RWKXedB1OvHUF84smGdmpkJpxOlZ6XQJEpX9FhX+vkgVS82rJ0UzdtESHl/D5ZZmaLWn7jfWBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ada032d5345d39ac8887d2b66f378547086bcf60d7ffd422fb96aa07c0ffea32","last_reissued_at":"2026-05-18T02:32:28.081689Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:28.081689Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A lower bound for the sum of the two largest signless Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Carla Oliveira, Leonardo de Lima","submitted_at":"2014-12-01T01:23:02Z","abstract_excerpt":"Let $G$ be a graph of order $n \\geq 3$ with sequence degree given as $d_{1}(G) \\geq ... \\geq d_{n}(G)$ and let $\\mu_1(G),..., \\mu_n(G)$ and $q_1(G), ..., q_{n}(G)$ be the Laplacian and signless Laplacian eigenvalues of $G$ arranged in non increasing order, respectively. Here, we consider the Grone's inequality [R. Grone, Eigenvalues and degree sequences of graphs, Lin. Multilin. Alg. 39 (1995) 133--136] $$ \\sum_{i=1}^{k} \\mu_{i}(G) \\geq \\sum_{i=1}^{k} d_{i}(G)+1$$ and prove that for $k=2$, the equality holds if and only if $G$ is the star graph $S_{n}.$ The signless Laplacian version of Grone'"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0323","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1412.0323","created_at":"2026-05-18T02:32:28.081750+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.0323v1","created_at":"2026-05-18T02:32:28.081750+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.0323","created_at":"2026-05-18T02:32:28.081750+00:00"},{"alias_kind":"pith_short_12","alias_value":"VWQDFVJULU42","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"VWQDFVJULU42ZCEH","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"VWQDFVJU","created_at":"2026-05-18T12:28:54.890064+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4","json":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4.json","graph_json":"https://pith.science/api/pith-number/VWQDFVJULU42ZCEH2K3G6N4FI4/graph.json","events_json":"https://pith.science/api/pith-number/VWQDFVJULU42ZCEH2K3G6N4FI4/events.json","paper":"https://pith.science/paper/VWQDFVJU"},"agent_actions":{"view_html":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4","download_json":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4.json","view_paper":"https://pith.science/paper/VWQDFVJU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.0323&json=true","fetch_graph":"https://pith.science/api/pith-number/VWQDFVJULU42ZCEH2K3G6N4FI4/graph.json","fetch_events":"https://pith.science/api/pith-number/VWQDFVJULU42ZCEH2K3G6N4FI4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4/action/storage_attestation","attest_author":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4/action/author_attestation","sign_citation":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4/action/citation_signature","submit_replication":"https://pith.science/pith/VWQDFVJULU42ZCEH2K3G6N4FI4/action/replication_record"}},"created_at":"2026-05-18T02:32:28.081750+00:00","updated_at":"2026-05-18T02:32:28.081750+00:00"}