{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:FZCR2U5WYXGIJ5BP4JWROTJIQ5","short_pith_number":"pith:FZCR2U5W","schema_version":"1.0","canonical_sha256":"2e451d53b6c5cc84f42fe26d174d28874979c9b934772a386b51968989a0a212","source":{"kind":"arxiv","id":"1710.02771","version":1},"attestation_state":"computed","paper":{"title":"Computing the $A_{\\alpha}-$ eigenvalues of a bug","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo","submitted_at":"2017-10-08T02:47:06Z","abstract_excerpt":"Let $G$ be a simple undirected graph. For $\\alpha \\in [0,1]$, let \\begin{equation*}\n  A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) , \\end{equation*} where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the degrees of $G$. In particular, $A_{0}(G)=A(G)$ and $A_{\\frac{1}{2}}(G)=\\frac{1}{2}Q(G)$ where $Q(G)$ is the signless Laplacian matrix of $G$. A bug $B_{p,q,r}$ is a graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. In \\cite{HaSt08}, Hansen and Stevanovi\\'{c} prove"},"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":"1710.02771","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-10-08T02:47:06Z","cross_cats_sorted":[],"title_canon_sha256":"d100384f1dd9afef3a192b8818214efc83ac6ba38fa50da101ea8b5baaf3a2d1","abstract_canon_sha256":"f7ced5c0a4b06eb1998b2874b75521be328d4d281949e90736ffda141c71c042"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:28.853650Z","signature_b64":"kaBeFugbZsGgOl0OEA6h5j1Qd4bEHZRJd7g0yPNzWvtU+ml4SFJz5JSehLVer5ywQUnZTSnsT0RRyVzMFoPNBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e451d53b6c5cc84f42fe26d174d28874979c9b934772a386b51968989a0a212","last_reissued_at":"2026-05-18T00:33:28.853040Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:28.853040Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Computing the $A_{\\alpha}-$ eigenvalues of a bug","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Oscar Rojo","submitted_at":"2017-10-08T02:47:06Z","abstract_excerpt":"Let $G$ be a simple undirected graph. For $\\alpha \\in [0,1]$, let \\begin{equation*}\n  A_{\\alpha}\\left( G\\right) =\\alpha D\\left( G\\right) +(1-\\alpha)A\\left( G\\right) , \\end{equation*} where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the diagonal matrix of the degrees of $G$. In particular, $A_{0}(G)=A(G)$ and $A_{\\frac{1}{2}}(G)=\\frac{1}{2}Q(G)$ where $Q(G)$ is the signless Laplacian matrix of $G$. A bug $B_{p,q,r}$ is a graph obtained from a complete graph $K_{p}$ by deleting an edge and attaching paths $P_{q}$ and $P_{r}$ to its ends. 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