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For any real numbers $p$, $q\\geq 1$, we define the $(p,q)$-spectral radius $\\lambda_{p,q}(G)$ as \\[ \\lambda_{p,q}(G):=\\max_{||{\\bf x}||_p=||{\\bf y}||_q=1} \\sum_{e\\in E(G)}\\Bigg(\\prod_{u\\in T(e)}x_u\\Bigg)\\Bigg(\\prod_{v\\in H(e)}y_v\\Bigg), \\] where ${\\bf x}=(x_1, \\ldots, x_m)^{{\\rm T}}$, ${\\bf y}=(y_1,\\ldots, y_n)^{{\\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. We study some properties abo"},"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":"1804.08808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-24T01:58:18Z","cross_cats_sorted":[],"title_canon_sha256":"7f4b5b2817f562d5b673386389b1e0162fe2b48401a0747ab885fdda5c380766","abstract_canon_sha256":"f7305d2d64cf8d64b802fde2274613cc72d2ab91ba69332572833304abcfe70f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:37.274829Z","signature_b64":"AJzxuLYefoedgeCQ4UJxu+oyLyyDpM9Qn5yUlqh3/qgIiF8nvIfI8D/mREfKbo/GvvOeT6uR761/YUogO1N8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"896b3bc7dac808a9ac53e9b6ba26eaba9e648ee5a31792e0e7c1816862d21cb0","last_reissued_at":"2026-05-18T00:17:37.274121Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:37.274121Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The $(p,q)$-spectral radii of $(r,s)$-directed hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lele Liu, Linyuan Lu","submitted_at":"2018-04-24T01:58:18Z","abstract_excerpt":"An $(r,s)$-directed hypergraph is a directed hypergraph with $r$ vertices in tail and $s$ vertices in head of each arc. Let $G$ be an $(r,s)$-directed hypergraph. For any real numbers $p$, $q\\geq 1$, we define the $(p,q)$-spectral radius $\\lambda_{p,q}(G)$ as \\[ \\lambda_{p,q}(G):=\\max_{||{\\bf x}||_p=||{\\bf y}||_q=1} \\sum_{e\\in E(G)}\\Bigg(\\prod_{u\\in T(e)}x_u\\Bigg)\\Bigg(\\prod_{v\\in H(e)}y_v\\Bigg), \\] where ${\\bf x}=(x_1, \\ldots, x_m)^{{\\rm T}}$, ${\\bf y}=(y_1,\\ldots, y_n)^{{\\rm T}}$ are real vectors; and $T(e)$, $H(e)$ are the tail and head of arc $e$, respectively. 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