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Discrete Math. 11 (2017) 81--107] defined the matrix $A_{\\alpha}(G)$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\\leq i\\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$"},"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":"1805.05808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-05-15T14:31:47Z","cross_cats_sorted":[],"title_canon_sha256":"6f67b7b67fd481c6b675b066fe507f1c960c92407d98a80d04c1721654561468","abstract_canon_sha256":"2fbe9ae9815ea5ff8301378abd9865301ea015d2252d84021772d9f606544773"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:15:53.618475Z","signature_b64":"syTYb9fUG426Rvdkla77DYPjMQju7rWxkmZPe8eSA137qXW8ANAlLrTSSt9o2qg5eXLK9g95I0aimsc2gyw+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a2450f1b7b75f8c28acb5427d1366320f587323f0b16e1bfed4bf49b2d28c676","last_reissued_at":"2026-05-18T00:15:53.617873Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:15:53.617873Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A note on the $A_{\\alpha}$-spectral radius of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiqiu Lin, Jie Xue, Xing Huang","submitted_at":"2018-05-15T14:31:47Z","abstract_excerpt":"Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\\alpha\\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017) 81--107] defined the matrix $A_{\\alpha}(G)$ as $A_{\\alpha}(G)=\\alpha D(G)+(1-\\alpha)A(G).$ Let $u$ and $v$ be two vertices of a connected graph $G$. Suppose that $u$ and $v$ are connected by a path $w_0(=v)w_1\\cdots w_{s-1}w_s(=u)$ where $d(w_i)=2$ for $1\\leq i\\leq s-1$. Let $G_{p,s,q}(u,v)$ be the graph obtained by attaching the paths $P_p$ to $u$ and $P_q$ to $v$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.05808","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":"1805.05808","created_at":"2026-05-18T00:15:53.617970+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.05808v1","created_at":"2026-05-18T00:15:53.617970+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.05808","created_at":"2026-05-18T00:15:53.617970+00:00"},{"alias_kind":"pith_short_12","alias_value":"UJCQ6G33OX4M","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"UJCQ6G33OX4MFCWL","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"UJCQ6G33","created_at":"2026-05-18T12:32:56.356000+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/UJCQ6G33OX4MFCWLKQT5CNTDED","json":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED.json","graph_json":"https://pith.science/api/pith-number/UJCQ6G33OX4MFCWLKQT5CNTDED/graph.json","events_json":"https://pith.science/api/pith-number/UJCQ6G33OX4MFCWLKQT5CNTDED/events.json","paper":"https://pith.science/paper/UJCQ6G33"},"agent_actions":{"view_html":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED","download_json":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED.json","view_paper":"https://pith.science/paper/UJCQ6G33","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.05808&json=true","fetch_graph":"https://pith.science/api/pith-number/UJCQ6G33OX4MFCWLKQT5CNTDED/graph.json","fetch_events":"https://pith.science/api/pith-number/UJCQ6G33OX4MFCWLKQT5CNTDED/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED/action/storage_attestation","attest_author":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED/action/author_attestation","sign_citation":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED/action/citation_signature","submit_replication":"https://pith.science/pith/UJCQ6G33OX4MFCWLKQT5CNTDED/action/replication_record"}},"created_at":"2026-05-18T00:15:53.617970+00:00","updated_at":"2026-05-18T00:15:53.617970+00:00"}