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Our main result is an explicit equation determining the number of `bumps' on paths in a graph: in a $d$-regular (not necessarily transitive) non-oriented graph let the series $G(t)$ count all paths between two fixed points weighted by their length $t^{length}$, and $F(u,t)$ count the same paths, weighted as $u^{number of bumps}t^{length}$. Then one has $$F(1-u,t)/(1-u^2t^2) = G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ We then derive the c"},"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":"math/0012161","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2000-12-18T04:01:02Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"091b646b9f842a41533d748c2db779dc414004313e4913a3d6062dcd48ca1ad4","abstract_canon_sha256":"b8691afa7a5198191af333178770727c98743435393200317cb71d8fdd7b5c13"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:12:00.294110Z","signature_b64":"zOSu5A00AF5eM8BCDt+1u2PgZR9nL1vSP+hoYhEDqJAYr0bi/m3cEqFigGhVysSquDGlNrU6aBSxDYB82of5CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21730fc3183aed164a767894b1000c0798cfab756ebd9ebc1fc7d209be5fe504","last_reissued_at":"2026-07-04T15:12:00.293748Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:12:00.293748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Counting Paths in Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Laurent Bartholdi","submitted_at":"2000-12-18T04:01:02Z","abstract_excerpt":"We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. 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Then one has $$F(1-u,t)/(1-u^2t^2) = G(t/(1+u(d-u)t^2))/(1+u(d-u)t^2).$$ We then derive the c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0012161","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0012161/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"math/0012161","created_at":"2026-07-04T15:12:00.293808+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0012161v2","created_at":"2026-07-04T15:12:00.293808+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0012161","created_at":"2026-07-04T15:12:00.293808+00:00"},{"alias_kind":"pith_short_12","alias_value":"EFZQ7QYYHLWR","created_at":"2026-07-04T15:12:00.293808+00:00"},{"alias_kind":"pith_short_16","alias_value":"EFZQ7QYYHLWRMSTW","created_at":"2026-07-04T15:12:00.293808+00:00"},{"alias_kind":"pith_short_8","alias_value":"EFZQ7QYY","created_at":"2026-07-04T15:12:00.293808+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/EFZQ7QYYHLWRMSTWPCKLCAAMA6","json":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6.json","graph_json":"https://pith.science/api/pith-number/EFZQ7QYYHLWRMSTWPCKLCAAMA6/graph.json","events_json":"https://pith.science/api/pith-number/EFZQ7QYYHLWRMSTWPCKLCAAMA6/events.json","paper":"https://pith.science/paper/EFZQ7QYY"},"agent_actions":{"view_html":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6","download_json":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6.json","view_paper":"https://pith.science/paper/EFZQ7QYY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0012161&json=true","fetch_graph":"https://pith.science/api/pith-number/EFZQ7QYYHLWRMSTWPCKLCAAMA6/graph.json","fetch_events":"https://pith.science/api/pith-number/EFZQ7QYYHLWRMSTWPCKLCAAMA6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6/action/storage_attestation","attest_author":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6/action/author_attestation","sign_citation":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6/action/citation_signature","submit_replication":"https://pith.science/pith/EFZQ7QYYHLWRMSTWPCKLCAAMA6/action/replication_record"}},"created_at":"2026-07-04T15:12:00.293808+00:00","updated_at":"2026-07-04T15:12:00.293808+00:00"}