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Let $0\\leq\\alpha\\leq1,$ and let \\[ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) \\] where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. A basic result on the $A_{\\alpha}-$ spectrum of $R\\{H\\}$ is obtained. This result is used to prove that if $H=B_{k}$ is a generalized Bethe tree on $k$ levels, then the eig"},"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":"1704.06730","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-22T01:07:01Z","cross_cats_sorted":[],"title_canon_sha256":"3b40bda872813283ee163e3b69591ae3839f769f3ad346bb70d6719b18c39ee6","abstract_canon_sha256":"11102e813a8a417fbd7cfde6cd412a3bb0a74262d6edbd9cf2ecdd4c02541708"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:57.752088Z","signature_b64":"DTw+onZh95JjQUqSP6OqNpydZ7ntaVTE9o39it8DInpC30fOqh2C1vdx6SeQ/AO/UqWor1GbHXhhFslWA9cmBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"873f9378e1f44b120d8f591c1405a4bbc875c6ce24b34562bf9af7b4232bf36e","last_reissued_at":"2026-05-18T00:45:57.751487Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:57.751487Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$A_{\\alpha}$-spectrum of a graph obtained by copies of a rooted graph and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo","submitted_at":"2017-04-22T01:07:01Z","abstract_excerpt":"Given a connected graph $R$ on $r$ vertices and a rooted graph $H,$ let $R\\{H\\}$ be the graph obtained from $r$ copies of $H$ and the graph $R$ by identifying the root of the $i-th$ copy of $H$ with the $i-th$ vertex of $R$. Let $0\\leq\\alpha\\leq1,$ and let \\[ A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G) \\] where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. A basic result on the $A_{\\alpha}-$ spectrum of $R\\{H\\}$ is obtained. This result is used to prove that if $H=B_{k}$ is a generalized Bethe tree on $k$ levels, then the eig"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06730","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":"1704.06730","created_at":"2026-05-18T00:45:57.751602+00:00"},{"alias_kind":"arxiv_version","alias_value":"1704.06730v1","created_at":"2026-05-18T00:45:57.751602+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06730","created_at":"2026-05-18T00:45:57.751602+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q47ZG6HB6RFR","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q47ZG6HB6RFREDMP","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q47ZG6HB","created_at":"2026-05-18T12:31:37.085036+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/Q47ZG6HB6RFREDMPLEOBIBNEXP","json":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP.json","graph_json":"https://pith.science/api/pith-number/Q47ZG6HB6RFREDMPLEOBIBNEXP/graph.json","events_json":"https://pith.science/api/pith-number/Q47ZG6HB6RFREDMPLEOBIBNEXP/events.json","paper":"https://pith.science/paper/Q47ZG6HB"},"agent_actions":{"view_html":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP","download_json":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP.json","view_paper":"https://pith.science/paper/Q47ZG6HB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1704.06730&json=true","fetch_graph":"https://pith.science/api/pith-number/Q47ZG6HB6RFREDMPLEOBIBNEXP/graph.json","fetch_events":"https://pith.science/api/pith-number/Q47ZG6HB6RFREDMPLEOBIBNEXP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP/action/storage_attestation","attest_author":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP/action/author_attestation","sign_citation":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP/action/citation_signature","submit_replication":"https://pith.science/pith/Q47ZG6HB6RFREDMPLEOBIBNEXP/action/replication_record"}},"created_at":"2026-05-18T00:45:57.751602+00:00","updated_at":"2026-05-18T00:45:57.751602+00:00"}