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Let $k(\\Gamma)$ be the diameter of $\\Gamma$ and denote $\\kappa(d,n)=\\min\\{k(\\Gamma):~\\textrm{ord}(\\Gamma)=n,\\textrm{deg}(\\Gamma)=d\\}$.\n  We give a closed expression, $\\ell(d,n)$, of a tight lower bound of $\\kappa(d,n)$ by using the so called {\\em solid density} introduced by Fiduccia, Forcade and Zito.\n  A digraph $\\Gamma$ of degree $d$ is called {\\em tight} when $k(\\Gamma)=\\kappa(d,|\\Gamma|)=\\ell(d,|\\Gam"},"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":"1806.03899","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-11T10:51:30Z","cross_cats_sorted":[],"title_canon_sha256":"12b44b2ed49b0ef1676f427d39cc962c81e09f227dcf768900aec25da569f917","abstract_canon_sha256":"d0b18ef4403b665440f880709fbc39f4f9cad9013aaf8023c9a61b7c5176ee82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:41.638209Z","signature_b64":"f8VflYpPKTi0sAY7RayQCXWcn/avGMpegsnm9MU8PrL3T7CVqN/oslVkudrFnTP2SuSFNpwzQO65Yl3k3V3gAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ae6b32a08b0042b3d1477671cfd695afbb0df1ab9bf93959c332cc1fe9a144e","last_reissued_at":"2026-05-18T00:13:41.637466Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:41.637466Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On solid density of Cayley digraphs on finite Abelian groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"F. Aguil\\'o, M. Zaragoz\\'a","submitted_at":"2018-06-11T10:51:30Z","abstract_excerpt":"Let $\\Gamma=$Cay$(G,T)$ be a Cayley digraph over a finite Abelian group $G$ with respect the generating set $T\\not\\ni0$. $\\Gamma$ has order ord$(\\Gamma)=|G|=n$ and degree deg$(\\Gamma)=|T|=d$. Let $k(\\Gamma)$ be the diameter of $\\Gamma$ and denote $\\kappa(d,n)=\\min\\{k(\\Gamma):~\\textrm{ord}(\\Gamma)=n,\\textrm{deg}(\\Gamma)=d\\}$.\n  We give a closed expression, $\\ell(d,n)$, of a tight lower bound of $\\kappa(d,n)$ by using the so called {\\em solid density} introduced by Fiduccia, Forcade and Zito.\n  A digraph $\\Gamma$ of degree $d$ is called {\\em tight} when $k(\\Gamma)=\\kappa(d,|\\Gamma|)=\\ell(d,|\\Gam"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.03899","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":"1806.03899","created_at":"2026-05-18T00:13:41.637594+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.03899v1","created_at":"2026-05-18T00:13:41.637594+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.03899","created_at":"2026-05-18T00:13:41.637594+00:00"},{"alias_kind":"pith_short_12","alias_value":"RLTLGKQIWACC","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_16","alias_value":"RLTLGKQIWACCWPIU","created_at":"2026-05-18T12:32:50.500415+00:00"},{"alias_kind":"pith_short_8","alias_value":"RLTLGKQI","created_at":"2026-05-18T12:32:50.500415+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/RLTLGKQIWACCWPIUO5TRZ7LJLL","json":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL.json","graph_json":"https://pith.science/api/pith-number/RLTLGKQIWACCWPIUO5TRZ7LJLL/graph.json","events_json":"https://pith.science/api/pith-number/RLTLGKQIWACCWPIUO5TRZ7LJLL/events.json","paper":"https://pith.science/paper/RLTLGKQI"},"agent_actions":{"view_html":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL","download_json":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL.json","view_paper":"https://pith.science/paper/RLTLGKQI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.03899&json=true","fetch_graph":"https://pith.science/api/pith-number/RLTLGKQIWACCWPIUO5TRZ7LJLL/graph.json","fetch_events":"https://pith.science/api/pith-number/RLTLGKQIWACCWPIUO5TRZ7LJLL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL/action/storage_attestation","attest_author":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL/action/author_attestation","sign_citation":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL/action/citation_signature","submit_replication":"https://pith.science/pith/RLTLGKQIWACCWPIUO5TRZ7LJLL/action/replication_record"}},"created_at":"2026-05-18T00:13:41.637594+00:00","updated_at":"2026-05-18T00:13:41.637594+00:00"}