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We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint union of complete bipartite graphs of the same size and isolated vertices, or a regular graph with four eigenvalues. Graphs from the first three families in this list are indeed strongly $\\ell$-w"},"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":"1208.3067","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-08-15T09:22:43Z","cross_cats_sorted":[],"title_canon_sha256":"00b50857cd23a5586adb07ed99b548152400459360297da62e93627d5138d21a","abstract_canon_sha256":"5f30bc6d5c8c3a921b7785fbbc8b5b5e9930bb93ed99dcc3cea4d35071d5fae8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:01.905537Z","signature_b64":"2CYHDnHeqkj2k/zTLCxMrJaYD2SsoCjpoJlR0AZ6WKo8Vz/y1RrOt4+Ncy2468mPLT5w6sKA3OLGhdFM0el3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7cd994138bcb4d1a71272888bd94dc88c6a8ac759420686d4094a86cc4c810f0","last_reissued_at":"2026-05-18T03:35:01.904798Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:01.904798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strongly walk-regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Edwin R. van Dam, Gholamreza Omidi","submitted_at":"2012-08-15T09:22:43Z","abstract_excerpt":"We study a generalization of strongly regular graphs. 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